Prove that `(sintheta-sin5theta+sin9theta-sin13theta)/(costheta-cos5theta-cos9theta+cos13theta)=cot4theta`

Prove that (1+cos theta)/(sin theta)=(1+sin theta+cos theta)/(1+sin theta-cos theta) .

Prove the following (sintheta-2sin^3theta) /(2cos^3theta-costheta) =tantheta

Prove the following (cos^2theta) /(1-tantheta) +sin^3theta/(sintheta-costheta) =1+sinthetacostheta

Prove that cos theta/(1-sin theta)=tan (pi/4+theta/2)

Prove the following sin^6theta+cos^6theta=1-3sin^2thetacos^2theta

Prove that (((1+cot theta +tan theta)(sintheta-costheta))/(sec^3theta -cosec^3theta)) = (sin^2theta cos^2theta)

Solve sin 2theta+sin 4 theta+sin 6 theta=0 .

Prove the following sin^4theta-cos^4theta=sin^2theta-cos^2theta=1-2cos^2theta=2sin^2theta-1

Prove that cos^2(pi/4-theta)-sin^2(pi/4-theta)=sin 2theta

Prove that 8 sin^4(theta/2)-8sin^2(theta/2)+1=cos2theta