Question
Prove the following identities:
$\frac{1}{(1+\sin\theta)}+\frac{1}{(1-\sin\theta)}=2\sec^2\theta$
$\frac{1}{(1+\sin\theta)}+\frac{1}{(1-\sin\theta)}=2\sec^2\theta$
✓
Answer
$\text{L.H.S.}=\frac{1}{(1+\sin\theta)}+\frac{1}{(1-\sin\theta)}$
$=\frac{(1-\sin\theta)+(1+\sin\theta)}{(1+\sin\theta)(1-\sin\theta)}$
$=\frac{2}{1-\sin^2\theta}$
$=\frac{2}{\cos^2\theta}$
$=2\sec^2\theta$
$=\text{R.H.S.}$
$=\frac{(1-\sin\theta)+(1+\sin\theta)}{(1+\sin\theta)(1-\sin\theta)}$
$=\frac{2}{1-\sin^2\theta}$
$=\frac{2}{\cos^2\theta}$
$=2\sec^2\theta$
$=\text{R.H.S.}$
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