227
Prove that $(\text{cosec} \theta + \sin \theta) (\text{cosec} \theta - \sin \theta) = \cot^2 \theta + \cos^2 \theta$.
Show SolutionHide Solution↓
(a) LHS = $$\begin{aligned}& (\text{cosec} \theta + \sin \theta) (\text{cosec} \theta - \sin \theta) \\ & = \frac{(1+\sin^2 \theta)(1-\sin^2 \theta)}{\sin^2 \theta} \\ & = (1 + \sin^2 \theta) (\frac{\cos^2 \theta}{\sin^2 \theta}) \\ & = (\cot^2 \theta + \cos^2 \theta)\end{aligned}$$