(sin 2'theta')/(1+cos 2'theta') × (cos 'theta')/(1+cos 'theta') = tan 'theta'/2 | Prove

Solution:

LHS

$= \dfrac{sin 2\theta}{1 + cos 2\theta} × \dfrac{cos \theta}{1 + cos\theta}$

$= \dfrac{sin 2\theta}{1 + 2cos^2 \theta - 1} × \dfrac{cos \theta}{1 + cos\theta}$

$= \dfrac{2sin \theta cos\theta}{2cos^2 \theta} × \dfrac{cos \theta}{1 + cos\theta}$

$= \dfrac{sin \theta}{cos \theta} × \dfrac{cos\theta}{1 + cos\theta}$

$= \dfrac{sin \theta}{1 + cos\theta}$

$= \dfrac{2 sin \frac{ \theta}{2} cos\frac{\theta}{2}}{1 + 2cos^2 \frac{\theta}{2} - 1}$

$= \dfrac{2 sin \frac{ \theta}{2} cos \frac{\theta}{2}}{2 cos^2 \frac{\theta}{2}}$

$= \dfrac{sin \frac{\theta}{2}}{cos \frac{\theta}{2}}$

$= tan \frac{\theta}{2}$

RHS