179
Prove that: $\sqrt{\frac{\csc \theta-1}{\csc \theta +1}} + \sqrt{\frac{\csc \theta +1}{\csc \theta-1}} = 2 \sec \theta$
Show SolutionHide Solution↓
LHS = $\frac{\sqrt{\csc \theta-1} \sqrt{\csc \theta-1} + \sqrt{\csc \theta+1} \sqrt{\csc \theta+1}}{\sqrt{(\csc \theta+1)(\csc \theta-1)}}$
$= \frac{\csc \theta-1 + \csc \theta+1}{\sqrt{\csc^2 \theta-1}}$
$= \frac{2 \csc \theta}{\sqrt{\cot^2 \theta}}$
$= \frac{2 \csc \theta}{\cot \theta}$
$= \frac{2/\sin \theta}{\cos \theta/\sin \theta} = \frac{2}{\cos \theta} = 2 \sec \theta = RHS$
$= \frac{\csc \theta-1 + \csc \theta+1}{\sqrt{\csc^2 \theta-1}}$
$= \frac{2 \csc \theta}{\sqrt{\cot^2 \theta}}$
$= \frac{2 \csc \theta}{\cot \theta}$
$= \frac{2/\sin \theta}{\cos \theta/\sin \theta} = \frac{2}{\cos \theta} = 2 \sec \theta = RHS$