Question

Prove:

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

Answer 1

$(\sin \theta +\text{cosec}\theta {)}^2+(\cos \theta +\text{sec}\theta {)}^2={\sin }^2\theta +{\text{cosec}}^2\theta +2\sin \theta \times \frac{1}{\sin \theta }+{\cos }^2\theta +{\text{sec}}^2\theta +2\cos \theta \times \frac{1}{\cos \theta }$

$=({\sin }^2\theta +{\cos }^2\theta )+1+{\cot }^2\theta +1{\tan }^2\theta +2+2=1+6+{\tan }^2\theta +{\cot }^2\theta =7+{\tan }^2\theta +{\cot }^2\theta$