Question

Prove that:(its urgent plz help someone who's intelligent enough !!)
$tan(2cot^-^1 \sqrt{ \frac{1-cosx}{1+cosx} } )+tanx$=0

Answer 1

$LHS=tan(2*Cot^{-1}\sqrt{\frac{1-Cos\ x}{1+Cos\ x}})+Tan\ x\\\\1-Cos\ x=2\ Sin^2(\frac{x}{2})\\\\1+Cos\ x=2\ Cos^2(\frac{x}{2})\\\\LHS=Tan(2*Cot^{-1}\sqrt{Tan^2(\frac{x}{2})})+Tan\ x\\\\We\ know\ Tan\frac{x}{2}= Cot(\frac{\pi}{2}-\frac{x}{2})\\\\LHS=Tan(2*(\frac{\pi-x}{2}))+Tan\ x\\\\=Tan(\pi-x)+Tan\ x=-Tan\ x + Tan\ x = 0$