Question
Prove the following identities:
$1+\frac{\tan^2\theta}{(1+\sec\theta)}=\sec\theta$
$1+\frac{\tan^2\theta}{(1+\sec\theta)}=\sec\theta$
✓
Answer
$\text{L.H.S.}=1+\frac{\tan^2\theta}{(1+\sec\theta)}=\frac{1+\sec\theta+\tan^2\theta}{(1+\sec\theta)}$
$=\frac{\sec^2\theta+\sec\theta}{(1+\sec\theta)}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta\Big]$
$=\frac{\sec\theta(1+\sec\theta)}{(1+\sec\theta)}=\sec\theta$
$=\text{R.H.S.}$
Hence, LHS = RHS.
$=\frac{\sec^2\theta+\sec\theta}{(1+\sec\theta)}$ $\Big[\because\big(1+\tan^2\theta\big)=\sec^2\theta\Big]$
$=\frac{\sec\theta(1+\sec\theta)}{(1+\sec\theta)}=\sec\theta$
$=\text{R.H.S.}$
Hence, LHS = RHS.
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