64 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



sin x i cos x 

 ~ i + cos x sin x 



+ tan 2 x - i 

 tan x 



3.11 Functions of the Sum and Difference of Two Angles. 



5.111 sin (x db y) = sin x cos y cos # sin y, 



= cos # cos y (tan # tan y), 



tan a; d= tan y . , . 



= - - - - sin (x =F y), 

 tan # =F tan y 



= - | cos (# + y) + cos (# y) > (tan # tan 



3.112 cos (x y) = cos x cos y =F sin x sin y, 



= cos x cos y (i =F tan x tan y), 



cot x =F tan 3; / . . x 

 = - * cos (# =F y). 

 cot * tan y 



cot =F tan x / . . \ 



Sm (^ rp y\ 



cot y tan a; =F i 

 = cos x sin y (cot y =F tan x). 



3.113 , N tan x tan 



tan (x y) = 



i =F tan a; tan y 



cot y =b cot a; 

 ~ cot x cot y =F i 



sin 2x sin 2y 



" COS 2X + COS 2y 

 - ^JU, p<\ "71A. 



3.114 , N cot * cot y =F i 



cot (x y) = - - - > 

 cot y cot x 



sin 20: =F sin iy 

 cos 2 of cos 2y 



3.115 The cosine and sine of the sum of any number of angles in terms of the 

 sine and cosine of the angles are given by the real and imaginary parts of 



cos (x t + x 2 +. . . . + z n ) + i sin (xi + #2 + . . . . + x n ) 



= (cos xi + i sin #1) (cos # 2 + * sin #2) ..... (cos # n + i sin #) 



