Question

Prove that: ${(\tan \theta +\csc \varphi )}^2-{\left(\cot \varphi -\sec \theta \right)}^2=2\tan \theta \cot \varphi \left(\csc \theta +\sec \varphi \right)$

Answer 1

$L.H.S.$ $=({\tan }^2\theta +{\csc }^2\varphi +2\tan \theta \csc \varphi )-({\cot }^2\varphi +{\sec }^2\theta -2\cot \varphi \sec \theta )$
$=({\csc }^2\varphi -{\cot }^2\varphi )-({\sec }^2\theta -{\tan }^2\theta )+2\tan \theta \cot \varphi (\frac{\csc \varphi }{\cot \varphi }+\frac{\sec \theta }{\tan \theta })$
$=1-1+2\tan \theta \cot \varphi [1/\cos \varphi +1/\sin \theta ]$
$=2\tan \theta \cot \varphi (\sec \varphi +\csc \theta )$.