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Prove that : $\frac{\tan \theta}{1+\tan^2\theta} + \frac{\cot \theta}{1+ \cot^2 \theta} = 2 \sin \theta \cos \theta$.
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L.H.S. $= \frac{\tan \theta}{\sec^2 \theta} + \frac{\cot \theta}{\cosec^2 \theta}$ (I) (1 Mark)
$= \frac{\sin \theta}{\cos \theta} \times \cos^2 \theta + \frac{\cos \theta}{\sin \theta} \times \sin^2 \theta$ (II) (1/2 Mark)
$= 2 \sin \theta \cos \theta = \text{R.H.S.}$ (III) (1/2 Mark)
$= \frac{\sin \theta}{\cos \theta} \times \cos^2 \theta + \frac{\cos \theta}{\sin \theta} \times \sin^2 \theta$ (II) (1/2 Mark)
$= 2 \sin \theta \cos \theta = \text{R.H.S.}$ (III) (1/2 Mark)