Page 696 - Handbook Of Integral Equations

Addition formulas

sin(x ± y) = sin x cos y ± cos x sin y, cos(x ± y) = cos x cos y ∓ sin x sin y,
tan x ± tan y 1 ∓ tan x tan y
tan(x ± y)= , cot(x ± y)= .
1 ∓ tan x tan y tan x ± tan y

Addition and subtraction of trigonometric functions
x + y x – y

sin x + sin y = 2 sin cos ,
2 2
x – y x + y

sin x – sin y = 2 sin cos ,
2 2
x + y x – y

cos x + cos y = 2 cos cos ,
2 2
x + y x – y

cos x – cos y = –2 sin sin ,
2 2
2
2
2
2
sin x – sin y = cos y – cos x = sin(x + y) sin(x – y),
2
2
sin x – cos y = – cos(x + y) cos(x – y),
sin(x ± y) sin(y ± x)
tan x ± tan y = , cot x ± cot y = ,
cos x cos y sin x sin y
a cos x + b sin x = r sin(x + ϕ)= r cos(x – ψ).
√
2
2
Here r = a + b , sin ϕ = a/r, cos ϕ = b/r, sin ψ = b/r, and cos ψ = a/r.
Products of trigonometric functions
1
sin x sin y = [cos(x – y) – cos(x + y)],
2
1
cos x cos y = [cos(x – y) + cos(x + y)],
2
1
sin x cos y = [sin(x – y) + sin(x + y)].
2
Powers of trigonometric functions
1
1
2
1
2
cos x = 1 cos 2x + , sin x = – cos 2x + ,
2 2 2 2
1
3
3
cos x = 1 cos 3x + 3 cos x, sin x = – sin 3x + 3 sin x,
4 4 4 4
3
3
4
4
cos x = 1 cos 4x + 1 cos 2x + , sin x = 1 cos 4x – 1 cos 2x + ,
8 2 8 8 2 8
5
5
cos x = 1 cos 5x + 5 cos 3x + 5 cos x, sin x = 1 sin 5x – 5 sin 3x + 5 sin x,
16 16 8 16 16 8
n–1
1 k 1 n
2n
cos x = C 2n cos[2(n – k)x]+ C ,
2n
2 2n–1 2 2n
k=0
n
1 k
2n+1
cos x = C 2n+1 cos[(2n – 2k +1)x],
2 2n
k=0
n–1
1 n–k k 1 n
2n
sin x = (–1) C 2n cos[2(n – k)x]+ C ,
2n
2 2n–1 2 2n
k=0
n
1 n–k k
2n+1
sin x = (–1) C 2n+1 sin[(2n – 2k +1)x].
2 2n
k=0
m!
k
Here C = are binomial coefficients (0! = 1).
m
k!(m – k)!
© 1998 by CRC Press LLC
© 1998 by CRC Press LLC
Page 680