`((3+cos^2x)(2+sin x))/((1-x)(1+x)(3+x)(5+x))>0`

Prove that , tan^(-1)((2 sin 2x)/(1+2 cos 2x))-(1)/(2)sin^(-1)((3 sin 2x)/(5+4 cos 2x))=x .

int(cos x)/([cos((x)/(2))+sin((x)/(2))]^(5))dx= 1) (2)/(3)[cosec((x)/(2))+sec((x)/(2))]^(3) 2) (2)/(3)[cos((x)/(2))+sin((x)/(2))]^(3)+C 3) (2)/(3)[sin((x)/(2))-cos((x)/(2))]^(3)+c 4) -(1)/(3)[sin((x)/(2))+cos((x)/(2))]^(-3)

Prove that, tan^(-1)((3 sin 2x)/(5+3 cos 2x))+tan^(-1)((1)/(4) tan x)=x .

Integrate the following: int{(5cos^3x+2sin^3 x)/(2sin^2 x*cos^2 x)+sqrt(1+sin2x)+(1+2sin x)/(cos^2x)+(1-cos2x)/(1+cos 2x)}dx

Integrate the following: int{(5cos^(3)x+2sin^(3)x)/(2sin^(2)x*cos^(2)x)+sqrt(1+sin2x)+(1+2sin x)/(cos^(2)x)+(1-cos2x)/(1+cos2x)}dx

(cos^(3) x- sin^(2) x)/(cos x - sin x)=(1)/(2) (2 + sin 2x)

lim_(x rarr0)((1-cos2x)sin5x)/(x^(2)sin3x)

If f_(n)(x) = (sin x)/(cos3x)+(sin 3x)/(cos 3^(2)x) +(sin 3^(2)x)/(cos 3^(3)x) +....+ (sin 3^(n-1)x)/(cos 3^(n)x)"Then" f_(2) ((pi)/(4)) + f_(3) ((pi)/(4))=