Question

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

Answer 1

We need to prove ${(\sin \theta +co\sec \theta )}^2+{(\cos \theta +\sec \theta )}^2=7+{\tan }^2\theta +{\cos }^2\theta$
L.H.S. $={(\sin \theta +co\sec \theta )}^2+{(\cos \theta +\sec \theta )}^2$

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

$=$ R.H.S.