Question
Prove the following.
$\frac{1}{1-\sin \theta}+\frac{1}{1+\sin \theta}=2 \sec ^2 \theta$
$\frac{1}{1-\sin \theta}+\frac{1}{1+\sin \theta}=2 \sec ^2 \theta$
✓
Answer
Taking LHS
$\frac{1}{1-\sin \theta}+\frac{1}{1+\sin \theta}$
$=\frac{1+\sin \theta+1-\sin \theta}{(1-\sin \theta)(1+\sin \theta)}$
$=\frac{2}{1-\sin ^2 \theta}$
$=\frac{2}{\cos ^2 \theta}\left[\sin ^2 \theta+\cos ^2 \theta=1\right]$
$=2 \sec ^2 \theta$
= RHS
= Proved
$\frac{1}{1-\sin \theta}+\frac{1}{1+\sin \theta}$
$=\frac{1+\sin \theta+1-\sin \theta}{(1-\sin \theta)(1+\sin \theta)}$
$=\frac{2}{1-\sin ^2 \theta}$
$=\frac{2}{\cos ^2 \theta}\left[\sin ^2 \theta+\cos ^2 \theta=1\right]$
$=2 \sec ^2 \theta$
= RHS
= Proved
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