Question

Prove that:
$\frac{1}{1+\sin \theta }+\frac{1}{1-\sin \theta }=2{\sec }^2\theta$

Answer 1

We have,
LHS = $\frac{1}{1+\sin \theta }+\frac{1}{1-\sin \theta }$

$\Rightarrow$ LHS = $\frac{1-\sin \theta +1+\sin \theta }{(1+\sin \theta )(1-\sin \theta )}$

$\Rightarrow$ LHS = $\frac{2}{1-{\sin }^2\theta }$

$\Rightarrow$ LHS = $\frac{2}{{\cos }^2\theta }$ $[\because 1-{\sin }^2\theta ={\cos }^2\theta ]$

$\Rightarrow$ LHS = $2se{c}^2\theta$ = RHS $[\because sec\theta =\frac{1}{\cos \theta }]$