Question

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

Answer 1

Let us consider $(\sin \theta +cosec\theta {)}^2+(\cos \theta +\sec \theta {)}^2$

$={\sin }^2\theta +cose{c}^2\theta +2\sin \theta cosec\theta +co{s}^2\theta +{\sec }^2\theta +2\cos \theta \sec \theta$

$={\sin }^2\theta +{\cos }^2\theta +cose{c}^2\theta +{\sec }^2\theta +2\sin \theta \frac{1}{\sin \theta }+2\cos \theta \frac{1}{\cos \theta }$

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

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