68 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 



3.176 sin 2x = 2 sin x cos x. 



sin 30; = sin #(3 - 4 sin 2 x) 



= sin x(4 cos 2 x i). 

 sin 4# = sin x(8 cos 3 x 4 cos x). 

 sin 5* = sin #(5-20 sin 2 x + 16 sin 4 #) 



= sin #(i6 cos 4 x 12 cos 2 x + i). 

 sin 6x = sin #(32 cos 5 # - 32 cos 3 x + 6 cos x). 



3.177 cos 2# = cos 2 x sin 2 # 



= 1 2 sin 2 # 



= 2 cos 2 x i. 

 cos 30; = cos #(4 cos 2 x 3) 



= cos x(i 4 sin 2 #). 

 cos 4X = 8 cos 4 x - 8 cos 2 # + i. 

 cos 52 = cos x(i6 cos 4 x 20 cos 2 x + 5) 



= cos x(i6 sin 4 # 12 sin 2 x + i). 

 cos 6x = 32 cos 6 # - 48 cos 4 x + 1 8 cos 2 x - i. 



O irro 2 tan 



3.178 tan 2x 



cot 20; = 



i - tan 2 # 



cot 2 x i 

 2 cot x 



3.180 Integral Powers of Sine and Cosine. 



3.181 n an even integer: 



sn" x = - cos nx-n cos ( - 2 )x + 



(^ - i) ( - 2) *,' w! 



~ - cos ^- 6 ....... " l8 



cos" * = 



-Lj cos + cos ( - 2 )ic + n(n ~ ^ cos (w - 4 )a? 



, n(n - i) ( - 2) 

 + - -: - cos ( - 6)* + . . . + 



