Question

Prove ${(\sin \theta +co\sec \theta )}^2+{(\cos \theta +\sec \theta )}^2=7+{\tan }^2\theta +{\cot }^2\theta$.

Answer 1

LHS $={(\sin \theta +co\sec \theta )}^2+{(\cos \theta +\sec \theta )}^2$
$={\sin }^2\theta +2\sin \theta co\sec \theta +co{\sec }^2\theta +{\cos }^2\theta +2\cos \theta \sin \theta +{\sec }^2\theta$
$={\sin }^2\theta +2co{\sec }^2\theta +{\sec }^2\theta =2+2+1+co{\sec }^2\theta +{\sec }^2\theta$
$=5+1+{\tan }^2\theta +1+{\cot }^2\theta$
$=7+{\tan }^2\theta +{\cot }^2\theta$