Question

Prove that : $(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$

Answer 1

To prove:

$(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2=7+{\tan }^2\theta +{\cot }^2\theta$

Solution:

L.H.S. $=(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\sec \theta {)}^2$

$={\sin }^2\theta +\text{cosec}{}^2\theta +2\sin \theta .\text{cosec}\theta +{\cos }^2\theta +{\sec }^2\theta +2\cos \theta .\sec \theta$

$=({\sin }^2\theta +{\cos }^2\theta )+1+{\cot }^2\theta +2+1+{\tan }^2\theta +2$

$=1+6+{\tan }^2\theta +{\cot }^2\theta$

$=7+{\tan }^2\theta +{\cot }^2\theta$

$=$ R.H.S.