Question

Prove: $(\tan +\cot \theta +\sec \theta )(\tan +\cot \theta -\sec \theta )=co{\sec }^2\theta$

Answer 1

$LHS=(\tan \theta +\cot \theta +\text{sec}\theta )(\tan \theta +\cot \theta -\text{sec}\theta )$

$=({\tan }^2\theta +{\cot }^2\theta {)}^2-{\text{sec}}^2\theta$

$={\tan }^2\theta +{\cot }^2\theta +2-{\text{sec}}^2\theta$

$=2-(1+{\text{cosec}}^2\theta )-({\sec }^2\theta -{\tan }^2\theta )={\text{cosec}}^2\theta =RHS$

$LHS=RHS$

Hence Proved