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