Prove that `cosx+cos2x+ … + cosnx=(sin(n+1/2)x-sin(x/2))/(2sin(x/2))` and hence prove that : `int_0^1(sin(n+1/2)x)/sin(x/2)dx=pi`

Prove that cos x+cos3x+cos5x+cos(2n-1)x=(sin2nx)/(2)sin x

Prove that: int_(0)^( pi/2)(sin x)/(sin x-cos x)dx=(pi)/(4)

int(cos x)/((1-sin x)(2-sin x))dx

If n is a positive integer, prove that: int_0^(2pi) (cos(n-1)x-cosnx)/(1-cosx)dx=2pi , hence or otherwise, show that int_0^(2pi) (sin((nx)/2)/sin(x/2))^2dx=2npi .

int_(0)^(pi//2)(cosx)/((1+sin^(2)x))dx=?

int_(0)^(pi//2) (cosx )/ (1+sin^(2)x)dx=

Prove that: int_(0)^(2 pi)(x sin^(2n)x)/(sin^(2n)+cos^(2n)x)dx=pi^(2)

int_0^pi (x sin x)/(1+sin x) dx=pi/2 (pi-2)

Prove that: ,,(sin x-sin3x)/(sin^(2)x-cos^(2)x)=2sin x

Prove that: sin(n+1)x sin(n+2)x+cos(n+1)x cos(n+2)x=c