Tue Apr 15 00:36:09 2008 run on win32
% 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(2tan(x)*(sec(x)^2 - tan(x)^2 - 1));
0
on rationalize;
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))
off rationalize;
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);
1
-----------------
sin(10*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((2tan(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_package 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-----------------------
on nopowers;
% for comparison with version 2.0
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
{-1,sin(x) + 1,cos(x),cos(x)}
trigfactorize(cos(x)**4-1, x);
2
{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);
***** TrigGCD/Factorize error: last arg must be [number*]variable.
trigfactorize(sin(x), 2x);
***** TrigGCD/Factorize error: 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
%-------------------------TrigGCD---------------------------
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+2a*b+cos(2x)-2a*sin(x)+2b*sin(x),
-1-2a*b+cos(2x)-2a*sin(x)-2b*sin(x), x/2);
x x
2*cos(---)*sin(---) + a
2 2
triggcd(sin(x)^10-1, cos(x), x);
cos(x)
triggcd(sin(5x)+sin(3x), cos(x), x);
cos(x)
triggcd(sin(3x)+sin(5x), sin(5x)+sin(7x), x);
2
sin(x)*(sin(x) - 1)
%-----------------------------------------------------------
% New facilities in version 2
%-----------------------------------------------------------
% TrigSimp applied to non-scalars data structures:
trigsimp( sin(2x) = cos(2x) );
2
2*cos(x)*sin(x)= - 2*sin(x) + 1
trigsimp( { sin(2x), cos(2x) } );
2
{2*cos(x)*sin(x), - 2*sin(x) + 1}
trigsimp( { sin(2x) = cos(2x) } );
2
{2*cos(x)*sin(x)= - 2*sin(x) + 1}
trigsimp( mat((sin(2x),cos(2x)),
(csc(2x),sec(2x))) );
[ 2 ]
[ 2*cos(x)*sin(x) - 2*sin(x) + 1]
[ ]
[ 1 - 1 ]
[----------------- --------------- ]
[ 2*cos(x)*sin(x) 2 ]
[ 2*sin(x) - 1 ]
% An amusing identify:
trigsimp(csc x - cot x - tan(x/2));
0
% which could be DERIVED like this:
trigsimp(csc x - cot x, x/2, tan);
x
tan(---)
2
% A silly illustration of multiple additional trig arguments:
trigsimp(csc x - cot x, x/2, x/3);
x 5 x 3 x
16*sin(---) - 24*sin(---) + 9*sin(---)
6 6 6
------------------------------------------------------------
x x 4 x x 2 x
16*cos(---)*sin(---) - 16*cos(---)*sin(---) + 3*cos(---)
6 6 6 6 6
% A more useful illustration of multiple additional trig arguments:
trigsimp(csc x - cot x + csc y - cot y, x/2, y/2, tan);
x y
tan(---) + tan(---)
2 2
%-----------------------------------------------------------
% New TrigFactorize facility:
off nopowers;
% REDUCE 3.7 default, gives more compact output ...
trigfactorize(sin(x)^2, x);
{{sin(x),2}}
trigfactorize(1+cos(x), x);
{{cos(x) + 1,1}}
trigfactorize(1+cos(x), x/2);
x
{{2,1},{cos(---),2}}
2
trigfactorize(1+cos(x), x/6);
x x 2
{{2,1},{cos(---),2},{ - 4*sin(---) + 1,2}}
6 6
trigfactorize(sin(x)*(1-cos(x)), x);
{{sin(x)*( - cos(x) + 1),1}}
trigfactorize(sin(x)*(1-cos(x)), x/2);
x x
{{4,1},{sin(---),3},{cos(---),1}}
2 2
trigfactorize(tan(x), x);
{{tan(x),1}}
trigfactorize(sin(3x), x);
2
{{sin(x),1},{ - 4*sin(x) + 3,1}}
trigfactorize(sin(4x) - 1, x);
2
{{-1,1},{2*cos(x)*sin(x) + 2*sin(x) - 1,2}}
trigfactorize(sin(x)^4 - 1, x);
2
{{-1,1},{cos(x),2},{sin(x) + 1,1}}
trigfactorize(cos(x)^4 - 1, x);
2
{{sin(x),2},{sin(x) - 2,1}}
trigfactorize(sin(x)^10 - cos(x)^6, x);
{{-1,1},
2 5
{cos(x)*sin(x) - cos(x) + sin(x) ,1},
2 5
{cos(x)*sin(x) - cos(x) - sin(x) ,1}}
trigfactorize(sin(x)*cos(y), x);
{{cos(y),1},{sin(x),1}}
trigfactorize(sin(2x)*cos(y)^2, y/2);
{{2*cos(x)*sin(x),1},
y y
{cos(---) - sin(---),2},
2 2
y y
{cos(---) + sin(---),2}}
2 2
trigfactorize(sin(y)^4 - x^2, y);
2 2
{{sin(y) + x,1},{sin(y) - x,1}}
trigfactorize(sin(x), x+1);
***** TrigGCD/Factorize error: last arg must be [number*]variable.
trigfactorize(sin(x), 2x);
***** TrigGCD/Factorize error: basis not possible.
trigfactorize(sin(x)*cosh(x), x/2);
{{2,1},
x x
{cosh(---) + i*sinh(---),1},
2 2
x x
{cosh(---) - i*sinh(---),1},
2 2
x
{sin(---),1},
2
x
{cos(---),1}}
2
trigfactorize(1 + cos(2x) + 2cos(x)*cosh(x), x/2);
{{4,1},
x x
{cos(---) - sin(---),1},
2 2
x x
{cos(---) + sin(---),1},
2 2
x x x x
{cos(---)*cosh(---) - i*sin(---)*sinh(---),
2 2 2 2
1},
x x x x
{cos(---)*cosh(---) + i*sin(---)*sinh(---),
2 2 2 2
1}}
end;
Time for test: 3492 ms, plus GC time: 85 ms