Question
Prove the following identites:
$\sin^2\theta+\frac{1}{\big(1+\tan^2\theta\big)}=1$
$\sin^2\theta+\frac{1}{\big(1+\tan^2\theta\big)}=1$
✓
Answer
$\text{L.H.S.}=\sin^2\theta+\frac{1}{1+\tan^2\theta}$
$=\sin^2\theta+\frac{1}{\sec^2\theta}$ $\begin{bmatrix}\because\big(1-\cos^2\theta\big)=\sin^2\theta\end{bmatrix}$
$=\sin^2\theta+\cos^2\theta$
$=1$
$=\text{R.H.S.}$
$\therefore\ \text{LHS}=\text{RHS}$
$=\sin^2\theta+\frac{1}{\sec^2\theta}$ $\begin{bmatrix}\because\big(1-\cos^2\theta\big)=\sin^2\theta\end{bmatrix}$
$=\sin^2\theta+\cos^2\theta$
$=1$
$=\text{R.H.S.}$
$\therefore\ \text{LHS}=\text{RHS}$
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