Question

Solve:
$\frac{\tan \theta }{(\sec \theta -1)}+\frac{\tan \theta }{(\sec \theta +1)}=2\csc \theta$

Answer 1

$L.H.S$ of equation;

$\begin{array}{c}\frac{\tan \theta }{\left(\sec \theta -1\right)}+\frac{\tan \theta }{\left(\sec \theta +1\right)}=\frac{\tan \theta \left(2\sec \theta \right)}{{\sec }^2\theta -1}\\ =\frac{2\sec \theta \tan \theta }{{\tan }^2\theta }\\ =2\cos ec\theta =R.H.S\end{array}$

Proved.