Trigonometry

$$\begin{array}{rll}
LHS&=&tan\theta (1+cos2\theta)\\\\
&=&\frac{sin\theta}{cos\theta}(1+cos^2\theta -sin^2\theta)\\\\
&=& \frac{sin\theta}{cos\theta}(cos^2\theta+cos^2\theta)\\\\
&=& \frac{sin\theta}{cos\theta}(2cos^2\theta)\\\\
&=& 2sin\theta cos\theta\\\\
&=& sin2\theta\\\\
&=& RHS
\end{array}$$

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