@@ -1,1153 +1,1153 @@ -REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... - - -% test file for trigsimp package -% -%---------------------trigsimp--------------------------- -trigsimp(tan(x+y),keepalltrig); - - - - (tan(x) + tan(y)) ----------------------- - tan(x)*tan(y) - 1 - -trigsimp(ws,keepalltrig,combine); - - -tan(x + y) - -trigsimp(sin(5x-9y)); - - - 4 9 4 7 - - 4096*cos(x)*sin(x) *sin(y) + 9216*cos(x)*sin(x) *sin(y) - - 4 5 4 3 - - 6912*cos(x)*sin(x) *sin(y) + 1920*cos(x)*sin(x) *sin(y) - - 4 2 9 - - 144*cos(x)*sin(x) *sin(y) + 3072*cos(x)*sin(x) *sin(y) - - 2 7 2 5 - - 6912*cos(x)*sin(x) *sin(y) + 5184*cos(x)*sin(x) *sin(y) - - 2 3 2 9 - - 1440*cos(x)*sin(x) *sin(y) + 108*cos(x)*sin(x) *sin(y) - 256*cos(x)*sin(y) - - 7 5 3 - + 576*cos(x)*sin(y) - 432*cos(x)*sin(y) + 120*cos(x)*sin(y) - - 5 8 5 6 - - 9*cos(x)*sin(y) + 4096*cos(y)*sin(x) *sin(y) - 7168*cos(y)*sin(x) *sin(y) - - 5 4 5 2 5 - + 3840*cos(y)*sin(x) *sin(y) - 640*cos(y)*sin(x) *sin(y) + 16*cos(y)*sin(x) - - 3 8 3 6 - - 5120*cos(y)*sin(x) *sin(y) + 8960*cos(y)*sin(x) *sin(y) - - 3 4 3 2 3 - - 4800*cos(y)*sin(x) *sin(y) + 800*cos(y)*sin(x) *sin(y) - 20*cos(y)*sin(x) - - 8 6 - + 1280*cos(y)*sin(x)*sin(y) - 2240*cos(y)*sin(x)*sin(y) - - 4 2 - + 1200*cos(y)*sin(x)*sin(y) - 200*cos(y)*sin(x)*sin(y) + 5*cos(y)*sin(x) - -trigsimp(ws,combine); - - -sin(5*x - 9*y) - -trigsimp(cos(10x),cos); - - - 10 8 6 4 2 -512*cos(x) - 1280*cos(x) + 1120*cos(x) - 400*cos(x) + 50*cos(x) - 1 - -trigsimp(cos(10x),sin); - - - 10 8 6 4 2 - - 512*sin(x) + 1280*sin(x) - 1120*sin(x) + 400*sin(x) - 50*sin(x) + 1 - -trigsimp((sin(x-a)+sin(x+a))/(cos(x-a) + cos(x+a))); - - - sin(x) --------- - cos(x) - -trigsimp(cos(6x+4y),sin); - - - 5 3 5 -256*cos(x)*cos(y)*sin(x) *sin(y) - 128*cos(x)*cos(y)*sin(x) *sin(y) - - 3 3 3 - - 256*cos(x)*cos(y)*sin(x) *sin(y) + 128*cos(x)*cos(y)*sin(x) *sin(y) - - 3 - + 48*cos(x)*cos(y)*sin(x)*sin(y) - 24*cos(x)*cos(y)*sin(x)*sin(y) - - 6 4 6 2 6 4 4 - - 256*sin(x) *sin(y) + 256*sin(x) *sin(y) - 32*sin(x) + 384*sin(x) *sin(y) - - 4 2 4 2 4 2 2 - - 384*sin(x) *sin(y) + 48*sin(x) - 144*sin(x) *sin(y) + 144*sin(x) *sin(y) - - 2 4 2 - - 18*sin(x) + 8*sin(y) - 8*sin(y) + 1 - -trigsimp(ws,expon); - - - 12*i*x + 8*i*y - e + 1 ---------------------- - 6*i*x + 4*i*y - 2*e - -trigsimp(ws,hyp); - - - 5 3 -256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) - - 5 - + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) - - 3 3 - + 256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) - - 3 - + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) - - 3 - + 48*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y) - - 6 4 - + 24*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y) + 256*sinh(i*x) *sinh(i*y) - - 6 2 6 4 4 - + 256*sinh(i*x) *sinh(i*y) + 32*sinh(i*x) + 384*sinh(i*x) *sinh(i*y) - - 4 2 4 2 4 - + 384*sinh(i*x) *sinh(i*y) + 48*sinh(i*x) + 144*sinh(i*x) *sinh(i*y) - - 2 2 2 4 2 - + 144*sinh(i*x) *sinh(i*y) + 18*sinh(i*x) + 8*sinh(i*y) + 8*sinh(i*y) + 1 - -trigsimp(ws,combine); - - -cosh(6*i*x + 4*i*y) - -trigsimp(ws,trig,combine); - - -cos(6*x + 4*y) - -trigsimp(sqrt(1-cos(2x))); - - -sqrt(2)*abs(sin(x)) - -trigsimp(sin(x)**20*cos(x)**20,sin); - - - 20 20 18 16 14 12 -sin(x) *(sin(x) - 10*sin(x) + 45*sin(x) - 120*sin(x) + 210*sin(x) - - 10 8 6 4 2 - - 252*sin(x) + 210*sin(x) - 120*sin(x) + 45*sin(x) - 10*sin(x) - - + 1) - -trigsimp(sin(x)**20*cos(x)**20,cos); - - - 20 20 18 16 14 12 -cos(x) *(cos(x) - 10*cos(x) + 45*cos(x) - 120*cos(x) + 210*cos(x) - - 10 8 6 4 2 - - 252*cos(x) + 210*cos(x) - 120*cos(x) + 45*cos(x) - 10*cos(x) - - + 1) - -trigsimp(sin(x)**20*cos(x)**20,compact); - - - 20 20 -cos(x) *sin(x) - -trigsimp(sin(x)**10,combine); - - - - cos(10*x) + 10*cos(8*x) - 45*cos(6*x) + 120*cos(4*x) - 210*cos(2*x) + 126 ------------------------------------------------------------------------------- - 512 - -trigsimp(ws,hyp); - - - 10 - - sinh(i*x) - -trigsimp(ws,expon); - - - 20*i*x 18*i*x 16*i*x 14*i*x 12*i*x 10*i*x -( - e + 10*e - 45*e + 120*e - 210*e + 252*e - - 8*i*x 6*i*x 4*i*x 2*i*x 10*i*x - - 210*e + 120*e - 45*e + 10*e - 1)/(1024*e ) - -trigsimp(ws,trig); - - - 10 -sin(x) - -int(sin(x+y)*cos(x-y)*tan(x),x); - - -int(cos(x - y)*sin(x + y)*tan(x),x) - -int(trigsimp(sin(x+y)*cos(x-y)*tan(x)),x); - - - 2 2 - cos(x) *x - cos(x)*sin(x) - 2*cos(y)*log(cos(x))*sin(y) + sin(x) *x ---------------------------------------------------------------------- - 2 - -% int(sin(x+y)*cos(x-y)/tan(x),x) hangs -int(trigsimp(sin(x+y)*cos(x-y)/tan(x)),x); - - - x 2 -(cos(x)*sin(x) - 2*cos(y)*log(tan(---) + 1)*sin(y) - 2 - - x - + 2*cos(y)*log(tan(---))*sin(y) + x)/2 - 2 - -trigsimp(2*tan(x)*(sec(x)**2 - tan(x)**2 - 1)); - - -0 - -df(sqrt(1+cos(x)),x,4); - - - 4 2 2 2 -(sqrt(cos(x) + 1)*( - 4*cos(x) - 20*cos(x) *sin(x) + 12*cos(x) - - 2 4 2 - - 4*cos(x)*sin(x) + 8*cos(x) - 15*sin(x) + 16*sin(x) ))/(16 - - 4 3 2 - *(cos(x) + 4*cos(x) + 6*cos(x) + 4*cos(x) + 1)) - -trigsimp(ws); - - - sqrt(cos(x) + 1) ------------------- - 16 - -df(2cos((x+y)/2)*cos((x-y)/2),x); - - - x - y x + y x + y x - y - - (cos(-------)*sin(-------) + cos(-------)*sin(-------)) - 2 2 2 2 - -trigsimp(ws,combine); - - - - sin(x) - -df(int(1/cos(x),x),x); - - - x 2 - - (tan(---) + 1) - 2 --------------------- - x 2 - tan(---) - 1 - 2 - -trigsimp(ws,combine); - - - 1 --------- - cos(x) - -trigsimp(cos(100x)); - - - 100 -633825300114114700748351602688*sin(x) - - 98 - - 15845632502852867518708790067200*sin(x) - - 96 - + 192128294097091018664344079564800*sin(x) - - 94 - - 1505335087771022414277335056384000*sin(x) - - 92 - + 8567473526884295537508113973248000*sin(x) - - 90 - - 37750993877408064336851542202122240*sin(x) - - 88 - + 134036108580690866727917044786790400*sin(x) - - 86 - - 394078512785625681900511864396185600*sin(x) - - 84 - + 978503372439851812055958467641344000*sin(x) - - 82 - - 2082455895192505138478065456775168000*sin(x) - - 80 - + 3842131126630171980492030767750184960*sin(x) - - 78 - - 6200783636440931286187342812099379200*sin(x) - - 76 - + 8816739233064449172547628060953804800*sin(x) - - 74 - - 11108623702136905456127648087408640000*sin(x) - - 72 - + 12460295938318846194767764735918080000*sin(x) - - 70 - - 12489614281703125832873100652943769600*sin(x) - - 68 - + 11221137831217652115471926367879168000*sin(x) - - 66 - - 9058026923994972189597820080095232000*sin(x) - - 64 - + 6581798018959761296303294062264320000*sin(x) - - 62 - - 4310885252184171141438414824407040000*sin(x) - - 60 - + 2547463753712583633893763260298035200*sin(x) - - 58 - - 1358954443662228159129584379363328000*sin(x) - - 56 - + 654531379770880870350000868032512000*sin(x) - - 54 - - 284578860769948204500000377405440000*sin(x) - - 52 - + 111631674825053695350740279623680000*sin(x) - - 50 - - 39472960218138986676021762874933248*sin(x) - - 48 - + 12566106098549963273941439584665600*sin(x) - - 46 - - 3595780740528756614156758967910400*sin(x) - - 44 - + 923024074019658505866132324352000*sin(x) - - 42 - - 212040013118649088525828358144000*sin(x) - - 40 - + 43468202689323063147794813419520*sin(x) - - 38 - - 7925478751208973645460484915200*sin(x) - - 36 - + 1280241627320751027747867852800*sin(x) - - 34 - - 182395347175955031090266112000*sin(x) - - 32 - + 22799418396994378886283264000*sin(x) - - 30 28 - - 2485387148331694929142087680*sin(x) + 234623135747458180159897600*sin(x) - - 26 24 - - 19023497493037149742694400*sin(x) + 1312104559685287280640000*sin(x) - - 22 20 - - 76111992112891822080000*sin(x) + 3662889620432918937600*sin(x) - - 18 16 - - 143850563845029888000*sin(x) + 4517474603507712000*sin(x) - - 14 12 - - 110586893598720000*sin(x) + 2042087523840000*sin(x) - - 10 8 6 - - 27227833651200*sin(x) + 246628928000*sin(x) - 1386112000*sin(x) - - 4 2 - + 4165000*sin(x) - 5000*sin(x) + 1 - -trigsimp(ws,combine); - - -cos(100*x) - -trigsimp(sinh(3a+4b-5c)*cosh(3a-5b-6c)); - - - 5 10 -16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 5 8 - + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 5 6 - + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 5 4 - + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 5 2 - + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 5 - + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a) - - 3 10 - + 16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 3 8 - + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 3 6 - + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 3 4 - + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 3 2 - + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) - - 3 - + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a) - - 10 - + 3072*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) - - 8 - + 6912*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) - - 6 - + 5376*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) - - 4 - + 1680*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) - - 2 - + 180*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) - - 5 11 - + 3*cosh(a)*cosh(b)*cosh(c)*sinh(a) + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 5 9 - + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 5 7 - + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 5 5 - + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 5 3 5 - + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c) + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 3 11 - + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 3 9 - + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 3 7 - + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 3 5 - + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 3 3 3 - + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c) + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c) - - 11 - + 3072*cosh(a)*sinh(a)*sinh(b)*sinh(c) - - 9 7 - + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c) + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c) - - 5 3 - + 3696*cosh(a)*sinh(a)*sinh(b)*sinh(c) + 660*cosh(a)*sinh(a)*sinh(b)*sinh(c) - - 6 11 - + 33*cosh(a)*sinh(a)*sinh(b)*sinh(c) - 16384*cosh(b)*sinh(a) *sinh(c) - - 6 9 6 7 - - 45056*cosh(b)*sinh(a) *sinh(c) - 45056*cosh(b)*sinh(a) *sinh(c) - - 6 5 6 3 - - 19712*cosh(b)*sinh(a) *sinh(c) - 3520*cosh(b)*sinh(a) *sinh(c) - - 6 4 11 - - 176*cosh(b)*sinh(a) *sinh(c) - 24576*cosh(b)*sinh(a) *sinh(c) - - 4 9 4 7 - - 67584*cosh(b)*sinh(a) *sinh(c) - 67584*cosh(b)*sinh(a) *sinh(c) - - 4 5 4 3 - - 29568*cosh(b)*sinh(a) *sinh(c) - 5280*cosh(b)*sinh(a) *sinh(c) - - 4 2 11 - - 264*cosh(b)*sinh(a) *sinh(c) - 9216*cosh(b)*sinh(a) *sinh(c) - - 2 9 2 7 - - 25344*cosh(b)*sinh(a) *sinh(c) - 25344*cosh(b)*sinh(a) *sinh(c) - - 2 5 2 3 - - 11088*cosh(b)*sinh(a) *sinh(c) - 1980*cosh(b)*sinh(a) *sinh(c) - - 2 8 - - 99*cosh(b)*sinh(a) *sinh(c) + 128*cosh(b)*sinh(b) *sinh(c) - - 6 4 - + 224*cosh(b)*sinh(b) *sinh(c) + 120*cosh(b)*sinh(b) *sinh(c) - - 2 11 9 - + 20*cosh(b)*sinh(b) *sinh(c) - 512*cosh(b)*sinh(c) - 1408*cosh(b)*sinh(c) - - 7 5 3 - - 1408*cosh(b)*sinh(c) - 616*cosh(b)*sinh(c) - 110*cosh(b)*sinh(c) - - 6 10 - - 5*cosh(b)*sinh(c) - 16384*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 6 8 - - 36864*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 6 6 - - 28672*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 6 4 - - 8960*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 6 2 6 - - 960*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 16*cosh(c)*sinh(a) *sinh(b) - - 4 10 - - 24576*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 4 8 - - 55296*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 4 6 - - 43008*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 4 4 - - 13440*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 4 2 4 - - 1440*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 24*cosh(c)*sinh(a) *sinh(b) - - 2 10 - - 9216*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 2 8 - - 20736*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 2 6 - - 16128*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 2 4 - - 5040*cosh(c)*sinh(a) *sinh(b)*sinh(c) - - 2 2 2 - - 540*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 9*cosh(c)*sinh(a) *sinh(b) - - 9 7 5 - + 128*cosh(c)*sinh(b) + 288*cosh(c)*sinh(b) + 216*cosh(c)*sinh(b) - - 3 10 - + 60*cosh(c)*sinh(b) - 512*cosh(c)*sinh(b)*sinh(c) - - 8 6 - - 1152*cosh(c)*sinh(b)*sinh(c) - 896*cosh(c)*sinh(b)*sinh(c) - - 4 2 - - 280*cosh(c)*sinh(b)*sinh(c) - 30*cosh(c)*sinh(b)*sinh(c) - - + 4*cosh(c)*sinh(b) - -trigsimp(ws,combine); - - - sinh(9*b + c) + sinh(6*a - b - 11*c) --------------------------------------- - 2 - -trigsimp(sec(20x-y),keepalltrig); - - - 20 20 20 19 -(csc(x) *csc(y)*sec(x) *sec(y))/(csc(x) *csc(y) + 20*csc(x) *sec(x)*sec(y) - - 18 2 17 3 - - 190*csc(x) *csc(y)*sec(x) - 1140*csc(x) *sec(x) *sec(y) - - 16 4 15 5 - + 4845*csc(x) *csc(y)*sec(x) + 15504*csc(x) *sec(x) *sec(y) - - 14 6 13 7 - - 38760*csc(x) *csc(y)*sec(x) - 77520*csc(x) *sec(x) *sec(y) - - 12 8 11 9 - + 125970*csc(x) *csc(y)*sec(x) + 167960*csc(x) *sec(x) *sec(y) - - 10 10 9 11 - - 184756*csc(x) *csc(y)*sec(x) - 167960*csc(x) *sec(x) *sec(y) - - 8 12 7 13 - + 125970*csc(x) *csc(y)*sec(x) + 77520*csc(x) *sec(x) *sec(y) - - 6 14 5 15 - - 38760*csc(x) *csc(y)*sec(x) - 15504*csc(x) *sec(x) *sec(y) - - 4 16 3 17 - + 4845*csc(x) *csc(y)*sec(x) + 1140*csc(x) *sec(x) *sec(y) - - 2 18 19 20 - - 190*csc(x) *csc(y)*sec(x) - 20*csc(x)*sec(x) *sec(y) + csc(y)*sec(x) ) - -trigsimp(csc(10a-9b),keepalltrig); - - - 10 9 10 9 10 8 -( - csc(a) *csc(b) *sec(a) *sec(b) )/(9*csc(a) *csc(b) *sec(b) - - 10 6 3 10 4 5 - - 84*csc(a) *csc(b) *sec(b) + 126*csc(a) *csc(b) *sec(b) - - 10 2 7 10 9 9 9 - - 36*csc(a) *csc(b) *sec(b) + csc(a) *sec(b) - 10*csc(a) *csc(b) *sec(a) - - 9 7 2 9 5 4 - + 360*csc(a) *csc(b) *sec(a)*sec(b) - 1260*csc(a) *csc(b) *sec(a)*sec(b) - - 9 3 6 9 8 - + 840*csc(a) *csc(b) *sec(a)*sec(b) - 90*csc(a) *csc(b)*sec(a)*sec(b) - - 8 8 2 8 6 2 3 - - 405*csc(a) *csc(b) *sec(a) *sec(b) + 3780*csc(a) *csc(b) *sec(a) *sec(b) - - 8 4 2 5 - - 5670*csc(a) *csc(b) *sec(a) *sec(b) - - 8 2 2 7 8 2 9 - + 1620*csc(a) *csc(b) *sec(a) *sec(b) - 45*csc(a) *sec(a) *sec(b) - - 7 9 3 7 7 3 2 - + 120*csc(a) *csc(b) *sec(a) - 4320*csc(a) *csc(b) *sec(a) *sec(b) - - 7 5 3 4 - + 15120*csc(a) *csc(b) *sec(a) *sec(b) - - 7 3 3 6 - - 10080*csc(a) *csc(b) *sec(a) *sec(b) - - 7 3 8 6 8 4 - + 1080*csc(a) *csc(b)*sec(a) *sec(b) + 1890*csc(a) *csc(b) *sec(a) *sec(b) - - 6 6 4 3 - - 17640*csc(a) *csc(b) *sec(a) *sec(b) - - 6 4 4 5 - + 26460*csc(a) *csc(b) *sec(a) *sec(b) - - 6 2 4 7 6 4 9 - - 7560*csc(a) *csc(b) *sec(a) *sec(b) + 210*csc(a) *sec(a) *sec(b) - - 5 9 5 5 7 5 2 - - 252*csc(a) *csc(b) *sec(a) + 9072*csc(a) *csc(b) *sec(a) *sec(b) - - 5 5 5 4 - - 31752*csc(a) *csc(b) *sec(a) *sec(b) - - 5 3 5 6 - + 21168*csc(a) *csc(b) *sec(a) *sec(b) - - 5 5 8 4 8 6 - - 2268*csc(a) *csc(b)*sec(a) *sec(b) - 1890*csc(a) *csc(b) *sec(a) *sec(b) - - 4 6 6 3 - + 17640*csc(a) *csc(b) *sec(a) *sec(b) - - 4 4 6 5 - - 26460*csc(a) *csc(b) *sec(a) *sec(b) - - 4 2 6 7 4 6 9 - + 7560*csc(a) *csc(b) *sec(a) *sec(b) - 210*csc(a) *sec(a) *sec(b) - - 3 9 7 3 7 7 2 - + 120*csc(a) *csc(b) *sec(a) - 4320*csc(a) *csc(b) *sec(a) *sec(b) - - 3 5 7 4 - + 15120*csc(a) *csc(b) *sec(a) *sec(b) - - 3 3 7 6 - - 10080*csc(a) *csc(b) *sec(a) *sec(b) - - 3 7 8 2 8 8 - + 1080*csc(a) *csc(b)*sec(a) *sec(b) + 405*csc(a) *csc(b) *sec(a) *sec(b) - - 2 6 8 3 - - 3780*csc(a) *csc(b) *sec(a) *sec(b) - - 2 4 8 5 - + 5670*csc(a) *csc(b) *sec(a) *sec(b) - - 2 2 8 7 2 8 9 - - 1620*csc(a) *csc(b) *sec(a) *sec(b) + 45*csc(a) *sec(a) *sec(b) - - 9 9 7 9 2 - - 10*csc(a)*csc(b) *sec(a) + 360*csc(a)*csc(b) *sec(a) *sec(b) - - 5 9 4 3 9 6 - - 1260*csc(a)*csc(b) *sec(a) *sec(b) + 840*csc(a)*csc(b) *sec(a) *sec(b) - - 9 8 8 10 - - 90*csc(a)*csc(b)*sec(a) *sec(b) - 9*csc(b) *sec(a) *sec(b) - - 6 10 3 4 10 5 - + 84*csc(b) *sec(a) *sec(b) - 126*csc(b) *sec(a) *sec(b) - - 2 10 7 10 9 - + 36*csc(b) *sec(a) *sec(b) - sec(a) *sec(b) ) - -trigsimp(ws,combine); - - - 2*cos(a) ----------------------------------- - sin(11*a - 9*b) + sin(9*a - 9*b) - -trigsimp(cosh(50*acosh(x))-cos(50*acos(x))); - - -0 - -trigsimp(cos(n*acos(x))-cosh(n*acosh(x)),trig); - - -0 - -trigsimp((2*tan(log(x))*(sec(log(x))**2 - tan(log(x))**2 - 1))/x); - - -0 - -trigsimp(sech(10x),keepalltrig); - - - 10 10 10 8 2 6 4 -(csch(x) *sech(x) )/(csch(x) + 45*csch(x) *sech(x) + 210*csch(x) *sech(x) - - 4 6 2 8 10 - + 210*csch(x) *sech(x) + 45*csch(x) *sech(x) + sech(x) ) - -trigsimp(ws,combine); - - - 1 ------------- - cosh(10*x) - -trigsimp(csch(3x-5y),keepalltrig); - - - 3 5 3 5 3 4 -( - csch(x) *csch(y) *sech(x) *sech(y) )/(5*csch(x) *csch(y) *sech(y) - - 3 2 3 3 5 - + 10*csch(x) *csch(y) *sech(y) + csch(x) *sech(y) - - 2 5 2 3 2 - - 3*csch(x) *csch(y) *sech(x) - 30*csch(x) *csch(y) *sech(x)*sech(y) - - 2 4 - - 15*csch(x) *csch(y)*sech(x)*sech(y) - - 4 2 - + 15*csch(x)*csch(y) *sech(x) *sech(y) - - 2 2 3 2 5 - + 30*csch(x)*csch(y) *sech(x) *sech(y) + 3*csch(x)*sech(x) *sech(y) - - 5 3 3 3 2 - - csch(y) *sech(x) - 10*csch(y) *sech(x) *sech(y) - - 3 4 - - 5*csch(y)*sech(x) *sech(y) ) - -trigsimp(ws,combine); - - - 1 ------------------ - sinh(3*x - 5*y) - -off precise; - - -trigsimp((sinh(x)+cosh(x))**n+(cosh(x)-sinh(x))**n,expon); - - - 2*n*x - e + 1 ------------- - n*x - e - -on precise; - - -trigsimp(ws,hyp); - - -2*cosh(n*x) - -load taylor; - - -taylor(sin(x+a)*cos(x+b),x,0,4); - - -cos(b)*sin(a) + (cos(a)*cos(b) - sin(a)*sin(b))*x - - 2 - - (cos(a)*sin(b) + cos(b)*sin(a))*x - - 2*( - cos(a)*cos(b) + sin(a)*sin(b)) 3 - + --------------------------------------*x - 3 - - cos(a)*sin(b) + cos(b)*sin(a) 4 5 - + -------------------------------*x + O(x ) - 3 - -trigsimp(ws,combine); - - - sin(a - b) + sin(a + b) 2 2*cos(a + b) 3 -------------------------- + cos(a + b)*x - sin(a + b)*x - --------------*x - 2 3 - - sin(a + b) 4 5 - + ------------*x + O(x ) - 3 - -%----------------------trigfactorize------------------------ -trigfactorize(sin(x)**2,x); - - -{sin(x),sin(x)} - -trigfactorize(1+cos(x),x); - - -{cos(x) + 1} - -trigfactorize(1+cos(x),x/2); - - - x x -{2,cos(---),cos(---)} - 2 2 - -trigfactorize(1+cos(x),x/6); - - -{2, - - x 2 - - 4*sin(---) + 1, - 6 - - x 2 - - 4*sin(---) + 1, - 6 - - x - cos(---), - 6 - - x - cos(---)} - 6 - -trigfactorize(sin(x)*(1-cos(x)),x); - - -{sin(x)*( - cos(x) + 1)} - -trigfactorize(sin(x)*(1-cos(x)),x/2); - - -{4, - - x - cos(---), - 2 - - x - sin(---), - 2 - - x - sin(---), - 2 - - x - sin(---)} - 2 - -trigfactorize(tan(x),x); - - -{tan(x)} - -trigfactorize(sin(x*3),x); - - - 2 -{ - 4*sin(x) + 3,sin(x)} - -trigfactorize(sin(4x)-1,x); - - -{-1, - - 2 - 2*cos(x)*sin(x) + 2*sin(x) - 1, - - 2 - 2*cos(x)*sin(x) + 2*sin(x) - 1} - -trigfactorize(sin(x)**4-1,x); - - - 2 -{ - (sin(x) + 1),cos(x),cos(x)} - -trigfactorize(cos(x)**4-1,x); - - - 2 -{-1, - sin(x) + 2,sin(x),sin(x)} - -trigfactorize(sin(x)**10-cos(x)**6,x); - - -{-1, - - 2 5 - cos(x)*sin(x) - cos(x) - sin(x) , - - 2 5 - cos(x)*sin(x) - cos(x) + sin(x) } - -trigfactorize(sin(x)*cos(y),x); - - -{cos(y),sin(x)} - -trigfactorize(sin(2x)*cos(y)**2,y/2); - - -{2*cos(x)*sin(x), - - y y - cos(---) + sin(---), - 2 2 - - y y - cos(---) + sin(---), - 2 2 - - y y - cos(---) - sin(---), - 2 2 - - y y - cos(---) - sin(---)} - 2 2 - -trigfactorize(sin(y)**4-x**2,y); - - - 2 2 -{ - sin(y) + x, - (sin(y) + x)} - -trigfactorize(sin(x),x+1); - - -***** error using trigfactorize, basis not possible - -trigfactorize(sin(x),2x); - - -***** error using trigfactorize, basis not possible - -trigfactorize(sin(x)*cosh(x),x/2); - - -{2, - - x - cos(---), - 2 - - x - sin(---), - 2 - - x x - cosh(---) - i*sinh(---), - 2 2 - - x x - cosh(---) + i*sinh(---)} - 2 2 - -trigfactorize(1+cos(2x)+2cos(x)*cosh(x),x/2); - - -{4, - - x x x x - cos(---)*cosh(---) + i*sin(---)*sinh(---), - 2 2 2 2 - - x x x x - cos(---)*cosh(---) - i*sin(---)*sinh(---), - 2 2 2 2 - - x x - cos(---) + sin(---), - 2 2 - - x x - cos(---) - sin(---)} - 2 2 - -%------------------------gcd--------------------------------- -triggcd(sin(x),cos(x),x); - - -1 - -triggcd(1-cos(x)**2,sin(x)**2,x); - - - 2 - - sin(x) - -triggcd(sin(x)**4-1,cos(x)**2,x); - - - 2 - - sin(x) + 1 - -triggcd(sin(5x+1),cos(x),x); - - -1 - -triggcd(1-cos(2x),sin(2x),x); - - -sin(x) - -triggcd(-5+cos(2x)-6sin(x),-7+cos(2x)-8sin(x),x/2); - - - x x -2*cos(---)*sin(---) + 1 - 2 2 - -triggcd(1-2cosh(x)+cosh(2x),1+2cosh(x)+cosh(2x),x/2); - - - x 2 -2*sinh(---) + 1 - 2 - -triggcd(1+cos(2x)+2cos(x)*cosh(x),1+2cos(x)*cosh(x)+cosh(2x),x/2); - - - x 2 x 2 - - sin(---) + sinh(---) + 1 - 2 2 - -triggcd(-1+2*a*b+cos(2x)-2*a*sin(x)+2*b*sin(x),-1-2*a*b+cos(2x)-2*a*sin(x) --2*b*sin(x),x/2); - - - x x -2*cos(---)*sin(---) + a - 2 2 - -triggcd(sin(x)**10-1,cos(x),x); - - -cos(x) - -triggcd(sin(5*x) + sin(3*x),cos(x),x); - - -cos(x) - -triggcd(sin(3x)+sin(5x),sin(5x)+sin(7x),x); - - - 2 -sin(x)*(sin(x) - 1) - - -end; -(TIME: trigsimp 75830 79750) +REDUCE 3.6, 15-Jul-95, patched to 6 Mar 96 ... + + +% test file for trigsimp package +% +%---------------------trigsimp--------------------------- +trigsimp(tan(x+y),keepalltrig); + + + - (tan(x) + tan(y)) +---------------------- + tan(x)*tan(y) - 1 + +trigsimp(ws,keepalltrig,combine); + + +tan(x + y) + +trigsimp(sin(5x-9y)); + + + 4 9 4 7 + - 4096*cos(x)*sin(x) *sin(y) + 9216*cos(x)*sin(x) *sin(y) + + 4 5 4 3 + - 6912*cos(x)*sin(x) *sin(y) + 1920*cos(x)*sin(x) *sin(y) + + 4 2 9 + - 144*cos(x)*sin(x) *sin(y) + 3072*cos(x)*sin(x) *sin(y) + + 2 7 2 5 + - 6912*cos(x)*sin(x) *sin(y) + 5184*cos(x)*sin(x) *sin(y) + + 2 3 2 9 + - 1440*cos(x)*sin(x) *sin(y) + 108*cos(x)*sin(x) *sin(y) - 256*cos(x)*sin(y) + + 7 5 3 + + 576*cos(x)*sin(y) - 432*cos(x)*sin(y) + 120*cos(x)*sin(y) + + 5 8 5 6 + - 9*cos(x)*sin(y) + 4096*cos(y)*sin(x) *sin(y) - 7168*cos(y)*sin(x) *sin(y) + + 5 4 5 2 5 + + 3840*cos(y)*sin(x) *sin(y) - 640*cos(y)*sin(x) *sin(y) + 16*cos(y)*sin(x) + + 3 8 3 6 + - 5120*cos(y)*sin(x) *sin(y) + 8960*cos(y)*sin(x) *sin(y) + + 3 4 3 2 3 + - 4800*cos(y)*sin(x) *sin(y) + 800*cos(y)*sin(x) *sin(y) - 20*cos(y)*sin(x) + + 8 6 + + 1280*cos(y)*sin(x)*sin(y) - 2240*cos(y)*sin(x)*sin(y) + + 4 2 + + 1200*cos(y)*sin(x)*sin(y) - 200*cos(y)*sin(x)*sin(y) + 5*cos(y)*sin(x) + +trigsimp(ws,combine); + + +sin(5*x - 9*y) + +trigsimp(cos(10x),cos); + + + 10 8 6 4 2 +512*cos(x) - 1280*cos(x) + 1120*cos(x) - 400*cos(x) + 50*cos(x) - 1 + +trigsimp(cos(10x),sin); + + + 10 8 6 4 2 + - 512*sin(x) + 1280*sin(x) - 1120*sin(x) + 400*sin(x) - 50*sin(x) + 1 + +trigsimp((sin(x-a)+sin(x+a))/(cos(x-a) + cos(x+a))); + + + sin(x) +-------- + cos(x) + +trigsimp(cos(6x+4y),sin); + + + 5 3 5 +256*cos(x)*cos(y)*sin(x) *sin(y) - 128*cos(x)*cos(y)*sin(x) *sin(y) + + 3 3 3 + - 256*cos(x)*cos(y)*sin(x) *sin(y) + 128*cos(x)*cos(y)*sin(x) *sin(y) + + 3 + + 48*cos(x)*cos(y)*sin(x)*sin(y) - 24*cos(x)*cos(y)*sin(x)*sin(y) + + 6 4 6 2 6 4 4 + - 256*sin(x) *sin(y) + 256*sin(x) *sin(y) - 32*sin(x) + 384*sin(x) *sin(y) + + 4 2 4 2 4 2 2 + - 384*sin(x) *sin(y) + 48*sin(x) - 144*sin(x) *sin(y) + 144*sin(x) *sin(y) + + 2 4 2 + - 18*sin(x) + 8*sin(y) - 8*sin(y) + 1 + +trigsimp(ws,expon); + + + 12*i*x + 8*i*y + e + 1 +--------------------- + 6*i*x + 4*i*y + 2*e + +trigsimp(ws,hyp); + + + 5 3 +256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) + + 5 + + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) + + 3 3 + + 256*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) + + 3 + + 128*cosh(i*x)*cosh(i*y)*sinh(i*x) *sinh(i*y) + + 3 + + 48*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y) + + 6 4 + + 24*cosh(i*x)*cosh(i*y)*sinh(i*x)*sinh(i*y) + 256*sinh(i*x) *sinh(i*y) + + 6 2 6 4 4 + + 256*sinh(i*x) *sinh(i*y) + 32*sinh(i*x) + 384*sinh(i*x) *sinh(i*y) + + 4 2 4 2 4 + + 384*sinh(i*x) *sinh(i*y) + 48*sinh(i*x) + 144*sinh(i*x) *sinh(i*y) + + 2 2 2 4 2 + + 144*sinh(i*x) *sinh(i*y) + 18*sinh(i*x) + 8*sinh(i*y) + 8*sinh(i*y) + 1 + +trigsimp(ws,combine); + + +cosh(6*i*x + 4*i*y) + +trigsimp(ws,trig,combine); + + +cos(6*x + 4*y) + +trigsimp(sqrt(1-cos(2x))); + + +sqrt(2)*abs(sin(x)) + +trigsimp(sin(x)**20*cos(x)**20,sin); + + + 20 20 18 16 14 12 +sin(x) *(sin(x) - 10*sin(x) + 45*sin(x) - 120*sin(x) + 210*sin(x) + + 10 8 6 4 2 + - 252*sin(x) + 210*sin(x) - 120*sin(x) + 45*sin(x) - 10*sin(x) + + + 1) + +trigsimp(sin(x)**20*cos(x)**20,cos); + + + 20 20 18 16 14 12 +cos(x) *(cos(x) - 10*cos(x) + 45*cos(x) - 120*cos(x) + 210*cos(x) + + 10 8 6 4 2 + - 252*cos(x) + 210*cos(x) - 120*cos(x) + 45*cos(x) - 10*cos(x) + + + 1) + +trigsimp(sin(x)**20*cos(x)**20,compact); + + + 20 20 +cos(x) *sin(x) + +trigsimp(sin(x)**10,combine); + + + - cos(10*x) + 10*cos(8*x) - 45*cos(6*x) + 120*cos(4*x) - 210*cos(2*x) + 126 +------------------------------------------------------------------------------ + 512 + +trigsimp(ws,hyp); + + + 10 + - sinh(i*x) + +trigsimp(ws,expon); + + + 20*i*x 18*i*x 16*i*x 14*i*x 12*i*x 10*i*x +( - e + 10*e - 45*e + 120*e - 210*e + 252*e + + 8*i*x 6*i*x 4*i*x 2*i*x 10*i*x + - 210*e + 120*e - 45*e + 10*e - 1)/(1024*e ) + +trigsimp(ws,trig); + + + 10 +sin(x) + +int(sin(x+y)*cos(x-y)*tan(x),x); + + +int(cos(x - y)*sin(x + y)*tan(x),x) + +int(trigsimp(sin(x+y)*cos(x-y)*tan(x)),x); + + + 2 2 + cos(x) *x - cos(x)*sin(x) - 2*cos(y)*log(cos(x))*sin(y) + sin(x) *x +--------------------------------------------------------------------- + 2 + +% int(sin(x+y)*cos(x-y)/tan(x),x) hangs +int(trigsimp(sin(x+y)*cos(x-y)/tan(x)),x); + + + x 2 +(cos(x)*sin(x) - 2*cos(y)*log(tan(---) + 1)*sin(y) + 2 + + x + + 2*cos(y)*log(tan(---))*sin(y) + x)/2 + 2 + +trigsimp(2*tan(x)*(sec(x)**2 - tan(x)**2 - 1)); + + +0 + +df(sqrt(1+cos(x)),x,4); + + + 4 2 2 2 +(sqrt(cos(x) + 1)*( - 4*cos(x) - 20*cos(x) *sin(x) + 12*cos(x) + + 2 4 2 + - 4*cos(x)*sin(x) + 8*cos(x) - 15*sin(x) + 16*sin(x) ))/(16 + + 4 3 2 + *(cos(x) + 4*cos(x) + 6*cos(x) + 4*cos(x) + 1)) + +trigsimp(ws); + + + sqrt(cos(x) + 1) +------------------ + 16 + +df(2cos((x+y)/2)*cos((x-y)/2),x); + + + x - y x + y x + y x - y + - (cos(-------)*sin(-------) + cos(-------)*sin(-------)) + 2 2 2 2 + +trigsimp(ws,combine); + + + - sin(x) + +df(int(1/cos(x),x),x); + + + x 2 + - (tan(---) + 1) + 2 +-------------------- + x 2 + tan(---) - 1 + 2 + +trigsimp(ws,combine); + + + 1 +-------- + cos(x) + +trigsimp(cos(100x)); + + + 100 +633825300114114700748351602688*sin(x) + + 98 + - 15845632502852867518708790067200*sin(x) + + 96 + + 192128294097091018664344079564800*sin(x) + + 94 + - 1505335087771022414277335056384000*sin(x) + + 92 + + 8567473526884295537508113973248000*sin(x) + + 90 + - 37750993877408064336851542202122240*sin(x) + + 88 + + 134036108580690866727917044786790400*sin(x) + + 86 + - 394078512785625681900511864396185600*sin(x) + + 84 + + 978503372439851812055958467641344000*sin(x) + + 82 + - 2082455895192505138478065456775168000*sin(x) + + 80 + + 3842131126630171980492030767750184960*sin(x) + + 78 + - 6200783636440931286187342812099379200*sin(x) + + 76 + + 8816739233064449172547628060953804800*sin(x) + + 74 + - 11108623702136905456127648087408640000*sin(x) + + 72 + + 12460295938318846194767764735918080000*sin(x) + + 70 + - 12489614281703125832873100652943769600*sin(x) + + 68 + + 11221137831217652115471926367879168000*sin(x) + + 66 + - 9058026923994972189597820080095232000*sin(x) + + 64 + + 6581798018959761296303294062264320000*sin(x) + + 62 + - 4310885252184171141438414824407040000*sin(x) + + 60 + + 2547463753712583633893763260298035200*sin(x) + + 58 + - 1358954443662228159129584379363328000*sin(x) + + 56 + + 654531379770880870350000868032512000*sin(x) + + 54 + - 284578860769948204500000377405440000*sin(x) + + 52 + + 111631674825053695350740279623680000*sin(x) + + 50 + - 39472960218138986676021762874933248*sin(x) + + 48 + + 12566106098549963273941439584665600*sin(x) + + 46 + - 3595780740528756614156758967910400*sin(x) + + 44 + + 923024074019658505866132324352000*sin(x) + + 42 + - 212040013118649088525828358144000*sin(x) + + 40 + + 43468202689323063147794813419520*sin(x) + + 38 + - 7925478751208973645460484915200*sin(x) + + 36 + + 1280241627320751027747867852800*sin(x) + + 34 + - 182395347175955031090266112000*sin(x) + + 32 + + 22799418396994378886283264000*sin(x) + + 30 28 + - 2485387148331694929142087680*sin(x) + 234623135747458180159897600*sin(x) + + 26 24 + - 19023497493037149742694400*sin(x) + 1312104559685287280640000*sin(x) + + 22 20 + - 76111992112891822080000*sin(x) + 3662889620432918937600*sin(x) + + 18 16 + - 143850563845029888000*sin(x) + 4517474603507712000*sin(x) + + 14 12 + - 110586893598720000*sin(x) + 2042087523840000*sin(x) + + 10 8 6 + - 27227833651200*sin(x) + 246628928000*sin(x) - 1386112000*sin(x) + + 4 2 + + 4165000*sin(x) - 5000*sin(x) + 1 + +trigsimp(ws,combine); + + +cos(100*x) + +trigsimp(sinh(3a+4b-5c)*cosh(3a-5b-6c)); + + + 5 10 +16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 5 8 + + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 5 6 + + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 5 4 + + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 5 2 + + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 5 + + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a) + + 3 10 + + 16384*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 3 8 + + 36864*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 3 6 + + 28672*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 3 4 + + 8960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 3 2 + + 960*cosh(a)*cosh(b)*cosh(c)*sinh(a) *sinh(c) + + 3 + + 16*cosh(a)*cosh(b)*cosh(c)*sinh(a) + + 10 + + 3072*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) + + 8 + + 6912*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) + + 6 + + 5376*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) + + 4 + + 1680*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) + + 2 + + 180*cosh(a)*cosh(b)*cosh(c)*sinh(a)*sinh(c) + + 5 11 + + 3*cosh(a)*cosh(b)*cosh(c)*sinh(a) + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 5 9 + + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 5 7 + + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 5 5 + + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 5 3 5 + + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c) + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 3 11 + + 16384*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 3 9 + + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 3 7 + + 45056*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 3 5 + + 19712*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 3 3 3 + + 3520*cosh(a)*sinh(a) *sinh(b)*sinh(c) + 176*cosh(a)*sinh(a) *sinh(b)*sinh(c) + + 11 + + 3072*cosh(a)*sinh(a)*sinh(b)*sinh(c) + + 9 7 + + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c) + 8448*cosh(a)*sinh(a)*sinh(b)*sinh(c) + + 5 3 + + 3696*cosh(a)*sinh(a)*sinh(b)*sinh(c) + 660*cosh(a)*sinh(a)*sinh(b)*sinh(c) + + 6 11 + + 33*cosh(a)*sinh(a)*sinh(b)*sinh(c) - 16384*cosh(b)*sinh(a) *sinh(c) + + 6 9 6 7 + - 45056*cosh(b)*sinh(a) *sinh(c) - 45056*cosh(b)*sinh(a) *sinh(c) + + 6 5 6 3 + - 19712*cosh(b)*sinh(a) *sinh(c) - 3520*cosh(b)*sinh(a) *sinh(c) + + 6 4 11 + - 176*cosh(b)*sinh(a) *sinh(c) - 24576*cosh(b)*sinh(a) *sinh(c) + + 4 9 4 7 + - 67584*cosh(b)*sinh(a) *sinh(c) - 67584*cosh(b)*sinh(a) *sinh(c) + + 4 5 4 3 + - 29568*cosh(b)*sinh(a) *sinh(c) - 5280*cosh(b)*sinh(a) *sinh(c) + + 4 2 11 + - 264*cosh(b)*sinh(a) *sinh(c) - 9216*cosh(b)*sinh(a) *sinh(c) + + 2 9 2 7 + - 25344*cosh(b)*sinh(a) *sinh(c) - 25344*cosh(b)*sinh(a) *sinh(c) + + 2 5 2 3 + - 11088*cosh(b)*sinh(a) *sinh(c) - 1980*cosh(b)*sinh(a) *sinh(c) + + 2 8 + - 99*cosh(b)*sinh(a) *sinh(c) + 128*cosh(b)*sinh(b) *sinh(c) + + 6 4 + + 224*cosh(b)*sinh(b) *sinh(c) + 120*cosh(b)*sinh(b) *sinh(c) + + 2 11 9 + + 20*cosh(b)*sinh(b) *sinh(c) - 512*cosh(b)*sinh(c) - 1408*cosh(b)*sinh(c) + + 7 5 3 + - 1408*cosh(b)*sinh(c) - 616*cosh(b)*sinh(c) - 110*cosh(b)*sinh(c) + + 6 10 + - 5*cosh(b)*sinh(c) - 16384*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 6 8 + - 36864*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 6 6 + - 28672*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 6 4 + - 8960*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 6 2 6 + - 960*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 16*cosh(c)*sinh(a) *sinh(b) + + 4 10 + - 24576*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 4 8 + - 55296*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 4 6 + - 43008*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 4 4 + - 13440*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 4 2 4 + - 1440*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 24*cosh(c)*sinh(a) *sinh(b) + + 2 10 + - 9216*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 2 8 + - 20736*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 2 6 + - 16128*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 2 4 + - 5040*cosh(c)*sinh(a) *sinh(b)*sinh(c) + + 2 2 2 + - 540*cosh(c)*sinh(a) *sinh(b)*sinh(c) - 9*cosh(c)*sinh(a) *sinh(b) + + 9 7 5 + + 128*cosh(c)*sinh(b) + 288*cosh(c)*sinh(b) + 216*cosh(c)*sinh(b) + + 3 10 + + 60*cosh(c)*sinh(b) - 512*cosh(c)*sinh(b)*sinh(c) + + 8 6 + - 1152*cosh(c)*sinh(b)*sinh(c) - 896*cosh(c)*sinh(b)*sinh(c) + + 4 2 + - 280*cosh(c)*sinh(b)*sinh(c) - 30*cosh(c)*sinh(b)*sinh(c) + + + 4*cosh(c)*sinh(b) + +trigsimp(ws,combine); + + + sinh(9*b + c) + sinh(6*a - b - 11*c) +-------------------------------------- + 2 + +trigsimp(sec(20x-y),keepalltrig); + + + 20 20 20 19 +(csc(x) *csc(y)*sec(x) *sec(y))/(csc(x) *csc(y) + 20*csc(x) *sec(x)*sec(y) + + 18 2 17 3 + - 190*csc(x) *csc(y)*sec(x) - 1140*csc(x) *sec(x) *sec(y) + + 16 4 15 5 + + 4845*csc(x) *csc(y)*sec(x) + 15504*csc(x) *sec(x) *sec(y) + + 14 6 13 7 + - 38760*csc(x) *csc(y)*sec(x) - 77520*csc(x) *sec(x) *sec(y) + + 12 8 11 9 + + 125970*csc(x) *csc(y)*sec(x) + 167960*csc(x) *sec(x) *sec(y) + + 10 10 9 11 + - 184756*csc(x) *csc(y)*sec(x) - 167960*csc(x) *sec(x) *sec(y) + + 8 12 7 13 + + 125970*csc(x) *csc(y)*sec(x) + 77520*csc(x) *sec(x) *sec(y) + + 6 14 5 15 + - 38760*csc(x) *csc(y)*sec(x) - 15504*csc(x) *sec(x) *sec(y) + + 4 16 3 17 + + 4845*csc(x) *csc(y)*sec(x) + 1140*csc(x) *sec(x) *sec(y) + + 2 18 19 20 + - 190*csc(x) *csc(y)*sec(x) - 20*csc(x)*sec(x) *sec(y) + csc(y)*sec(x) ) + +trigsimp(csc(10a-9b),keepalltrig); + + + 10 9 10 9 10 8 +( - csc(a) *csc(b) *sec(a) *sec(b) )/(9*csc(a) *csc(b) *sec(b) + + 10 6 3 10 4 5 + - 84*csc(a) *csc(b) *sec(b) + 126*csc(a) *csc(b) *sec(b) + + 10 2 7 10 9 9 9 + - 36*csc(a) *csc(b) *sec(b) + csc(a) *sec(b) - 10*csc(a) *csc(b) *sec(a) + + 9 7 2 9 5 4 + + 360*csc(a) *csc(b) *sec(a)*sec(b) - 1260*csc(a) *csc(b) *sec(a)*sec(b) + + 9 3 6 9 8 + + 840*csc(a) *csc(b) *sec(a)*sec(b) - 90*csc(a) *csc(b)*sec(a)*sec(b) + + 8 8 2 8 6 2 3 + - 405*csc(a) *csc(b) *sec(a) *sec(b) + 3780*csc(a) *csc(b) *sec(a) *sec(b) + + 8 4 2 5 + - 5670*csc(a) *csc(b) *sec(a) *sec(b) + + 8 2 2 7 8 2 9 + + 1620*csc(a) *csc(b) *sec(a) *sec(b) - 45*csc(a) *sec(a) *sec(b) + + 7 9 3 7 7 3 2 + + 120*csc(a) *csc(b) *sec(a) - 4320*csc(a) *csc(b) *sec(a) *sec(b) + + 7 5 3 4 + + 15120*csc(a) *csc(b) *sec(a) *sec(b) + + 7 3 3 6 + - 10080*csc(a) *csc(b) *sec(a) *sec(b) + + 7 3 8 6 8 4 + + 1080*csc(a) *csc(b)*sec(a) *sec(b) + 1890*csc(a) *csc(b) *sec(a) *sec(b) + + 6 6 4 3 + - 17640*csc(a) *csc(b) *sec(a) *sec(b) + + 6 4 4 5 + + 26460*csc(a) *csc(b) *sec(a) *sec(b) + + 6 2 4 7 6 4 9 + - 7560*csc(a) *csc(b) *sec(a) *sec(b) + 210*csc(a) *sec(a) *sec(b) + + 5 9 5 5 7 5 2 + - 252*csc(a) *csc(b) *sec(a) + 9072*csc(a) *csc(b) *sec(a) *sec(b) + + 5 5 5 4 + - 31752*csc(a) *csc(b) *sec(a) *sec(b) + + 5 3 5 6 + + 21168*csc(a) *csc(b) *sec(a) *sec(b) + + 5 5 8 4 8 6 + - 2268*csc(a) *csc(b)*sec(a) *sec(b) - 1890*csc(a) *csc(b) *sec(a) *sec(b) + + 4 6 6 3 + + 17640*csc(a) *csc(b) *sec(a) *sec(b) + + 4 4 6 5 + - 26460*csc(a) *csc(b) *sec(a) *sec(b) + + 4 2 6 7 4 6 9 + + 7560*csc(a) *csc(b) *sec(a) *sec(b) - 210*csc(a) *sec(a) *sec(b) + + 3 9 7 3 7 7 2 + + 120*csc(a) *csc(b) *sec(a) - 4320*csc(a) *csc(b) *sec(a) *sec(b) + + 3 5 7 4 + + 15120*csc(a) *csc(b) *sec(a) *sec(b) + + 3 3 7 6 + - 10080*csc(a) *csc(b) *sec(a) *sec(b) + + 3 7 8 2 8 8 + + 1080*csc(a) *csc(b)*sec(a) *sec(b) + 405*csc(a) *csc(b) *sec(a) *sec(b) + + 2 6 8 3 + - 3780*csc(a) *csc(b) *sec(a) *sec(b) + + 2 4 8 5 + + 5670*csc(a) *csc(b) *sec(a) *sec(b) + + 2 2 8 7 2 8 9 + - 1620*csc(a) *csc(b) *sec(a) *sec(b) + 45*csc(a) *sec(a) *sec(b) + + 9 9 7 9 2 + - 10*csc(a)*csc(b) *sec(a) + 360*csc(a)*csc(b) *sec(a) *sec(b) + + 5 9 4 3 9 6 + - 1260*csc(a)*csc(b) *sec(a) *sec(b) + 840*csc(a)*csc(b) *sec(a) *sec(b) + + 9 8 8 10 + - 90*csc(a)*csc(b)*sec(a) *sec(b) - 9*csc(b) *sec(a) *sec(b) + + 6 10 3 4 10 5 + + 84*csc(b) *sec(a) *sec(b) - 126*csc(b) *sec(a) *sec(b) + + 2 10 7 10 9 + + 36*csc(b) *sec(a) *sec(b) - sec(a) *sec(b) ) + +trigsimp(ws,combine); + + + 2*cos(a) +---------------------------------- + sin(11*a - 9*b) + sin(9*a - 9*b) + +trigsimp(cosh(50*acosh(x))-cos(50*acos(x))); + + +0 + +trigsimp(cos(n*acos(x))-cosh(n*acosh(x)),trig); + + +0 + +trigsimp((2*tan(log(x))*(sec(log(x))**2 - tan(log(x))**2 - 1))/x); + + +0 + +trigsimp(sech(10x),keepalltrig); + + + 10 10 10 8 2 6 4 +(csch(x) *sech(x) )/(csch(x) + 45*csch(x) *sech(x) + 210*csch(x) *sech(x) + + 4 6 2 8 10 + + 210*csch(x) *sech(x) + 45*csch(x) *sech(x) + sech(x) ) + +trigsimp(ws,combine); + + + 1 +------------ + cosh(10*x) + +trigsimp(csch(3x-5y),keepalltrig); + + + 3 5 3 5 3 4 +( - csch(x) *csch(y) *sech(x) *sech(y) )/(5*csch(x) *csch(y) *sech(y) + + 3 2 3 3 5 + + 10*csch(x) *csch(y) *sech(y) + csch(x) *sech(y) + + 2 5 2 3 2 + - 3*csch(x) *csch(y) *sech(x) - 30*csch(x) *csch(y) *sech(x)*sech(y) + + 2 4 + - 15*csch(x) *csch(y)*sech(x)*sech(y) + + 4 2 + + 15*csch(x)*csch(y) *sech(x) *sech(y) + + 2 2 3 2 5 + + 30*csch(x)*csch(y) *sech(x) *sech(y) + 3*csch(x)*sech(x) *sech(y) + + 5 3 3 3 2 + - csch(y) *sech(x) - 10*csch(y) *sech(x) *sech(y) + + 3 4 + - 5*csch(y)*sech(x) *sech(y) ) + +trigsimp(ws,combine); + + + 1 +----------------- + sinh(3*x - 5*y) + +off precise; + + +trigsimp((sinh(x)+cosh(x))**n+(cosh(x)-sinh(x))**n,expon); + + + 2*n*x + e + 1 +------------ + n*x + e + +on precise; + + +trigsimp(ws,hyp); + + +2*cosh(n*x) + +load taylor; + + +taylor(sin(x+a)*cos(x+b),x,0,4); + + +cos(b)*sin(a) + (cos(a)*cos(b) - sin(a)*sin(b))*x + + 2 + - (cos(a)*sin(b) + cos(b)*sin(a))*x + + 2*( - cos(a)*cos(b) + sin(a)*sin(b)) 3 + + --------------------------------------*x + 3 + + cos(a)*sin(b) + cos(b)*sin(a) 4 5 + + -------------------------------*x + O(x ) + 3 + +trigsimp(ws,combine); + + + sin(a - b) + sin(a + b) 2 2*cos(a + b) 3 +------------------------- + cos(a + b)*x - sin(a + b)*x - --------------*x + 2 3 + + sin(a + b) 4 5 + + ------------*x + O(x ) + 3 + +%----------------------trigfactorize------------------------ +trigfactorize(sin(x)**2,x); + + +{sin(x),sin(x)} + +trigfactorize(1+cos(x),x); + + +{cos(x) + 1} + +trigfactorize(1+cos(x),x/2); + + + x x +{2,cos(---),cos(---)} + 2 2 + +trigfactorize(1+cos(x),x/6); + + +{2, + + x 2 + - 4*sin(---) + 1, + 6 + + x 2 + - 4*sin(---) + 1, + 6 + + x + cos(---), + 6 + + x + cos(---)} + 6 + +trigfactorize(sin(x)*(1-cos(x)),x); + + +{sin(x)*( - cos(x) + 1)} + +trigfactorize(sin(x)*(1-cos(x)),x/2); + + +{4, + + x + cos(---), + 2 + + x + sin(---), + 2 + + x + sin(---), + 2 + + x + sin(---)} + 2 + +trigfactorize(tan(x),x); + + +{tan(x)} + +trigfactorize(sin(x*3),x); + + + 2 +{ - 4*sin(x) + 3,sin(x)} + +trigfactorize(sin(4x)-1,x); + + +{-1, + + 2 + 2*cos(x)*sin(x) + 2*sin(x) - 1, + + 2 + 2*cos(x)*sin(x) + 2*sin(x) - 1} + +trigfactorize(sin(x)**4-1,x); + + + 2 +{ - (sin(x) + 1),cos(x),cos(x)} + +trigfactorize(cos(x)**4-1,x); + + + 2 +{-1, - sin(x) + 2,sin(x),sin(x)} + +trigfactorize(sin(x)**10-cos(x)**6,x); + + +{-1, + + 2 5 + cos(x)*sin(x) - cos(x) - sin(x) , + + 2 5 + cos(x)*sin(x) - cos(x) + sin(x) } + +trigfactorize(sin(x)*cos(y),x); + + +{cos(y),sin(x)} + +trigfactorize(sin(2x)*cos(y)**2,y/2); + + +{2*cos(x)*sin(x), + + y y + cos(---) + sin(---), + 2 2 + + y y + cos(---) + sin(---), + 2 2 + + y y + cos(---) - sin(---), + 2 2 + + y y + cos(---) - sin(---)} + 2 2 + +trigfactorize(sin(y)**4-x**2,y); + + + 2 2 +{ - sin(y) + x, - (sin(y) + x)} + +trigfactorize(sin(x),x+1); + + +***** error using trigfactorize, basis not possible + +trigfactorize(sin(x),2x); + + +***** error using trigfactorize, basis not possible + +trigfactorize(sin(x)*cosh(x),x/2); + + +{2, + + x + cos(---), + 2 + + x + sin(---), + 2 + + x x + cosh(---) - i*sinh(---), + 2 2 + + x x + cosh(---) + i*sinh(---)} + 2 2 + +trigfactorize(1+cos(2x)+2cos(x)*cosh(x),x/2); + + +{4, + + x x x x + cos(---)*cosh(---) + i*sin(---)*sinh(---), + 2 2 2 2 + + x x x x + cos(---)*cosh(---) - i*sin(---)*sinh(---), + 2 2 2 2 + + x x + cos(---) + sin(---), + 2 2 + + x x + cos(---) - sin(---)} + 2 2 + +%------------------------gcd--------------------------------- +triggcd(sin(x),cos(x),x); + + +1 + +triggcd(1-cos(x)**2,sin(x)**2,x); + + + 2 + - sin(x) + +triggcd(sin(x)**4-1,cos(x)**2,x); + + + 2 + - sin(x) + 1 + +triggcd(sin(5x+1),cos(x),x); + + +1 + +triggcd(1-cos(2x),sin(2x),x); + + +sin(x) + +triggcd(-5+cos(2x)-6sin(x),-7+cos(2x)-8sin(x),x/2); + + + x x +2*cos(---)*sin(---) + 1 + 2 2 + +triggcd(1-2cosh(x)+cosh(2x),1+2cosh(x)+cosh(2x),x/2); + + + x 2 +2*sinh(---) + 1 + 2 + +triggcd(1+cos(2x)+2cos(x)*cosh(x),1+2cos(x)*cosh(x)+cosh(2x),x/2); + + + x 2 x 2 + - sin(---) + sinh(---) + 1 + 2 2 + +triggcd(-1+2*a*b+cos(2x)-2*a*sin(x)+2*b*sin(x),-1-2*a*b+cos(2x)-2*a*sin(x) +-2*b*sin(x),x/2); + + + x x +2*cos(---)*sin(---) + a + 2 2 + +triggcd(sin(x)**10-1,cos(x),x); + + +cos(x) + +triggcd(sin(5*x) + sin(3*x),cos(x),x); + + +cos(x) + +triggcd(sin(3x)+sin(5x),sin(5x)+sin(7x),x); + + + 2 +sin(x)*(sin(x) - 1) + + +end; +(TIME: trigsimp 75830 79750)