Prove that `cos^(-1)((cosx+cosy)/(1+cosxcosy))=2tan^(-1)(tan(x/2)tan(y/2))`

Prove that (sinx-siny)/(cosx+cosy)=tan((x-y)/(2))

Prove that (sinx+sin3x)/(cosx+cos3x)=tan2x

Prove that : (cosx+cosy)^(2)+(sinx-siny)^(2)=4cos^(2)""(x+y)/(2)

Prove that, tan^(-1)x+"tan"^(-1)(2x)/(1-x^(2))=tan^(-1)((3x-x^(3))/(1-3x^(2))),|x|lt1/sqrt3

Prove the followings : If tan^(-1)((yz)/(xr))+tan^(-1)((zx)/(yr))+tan^(-1)((xy)/(zr))=pi/2 then x^(2)+y^(2)+z^(2)=r^(2) .

Solve the following equations : tan^(-1)((1-x)/(1+x))=1/2tan^(-1)x

Prove that sec^2(tan^(-1)2)+cos e c^2(cot^(-1)3)=15

Solve the equation tan^(-1) ((x-1)/(x-2))+tan^(-1)((x+1)/(x+2))=pi/4

Solve : 2tan^(-1)(cosx)=tan^(-1)(2cosecx) .