Question

Prove the following identities
$\frac{sin2\alpha }{1+cos2\alpha }\cdot \frac{cos\alpha }{1+cos\alpha }=tan\frac{\alpha }{2}$.

Answer 1

$LHS=\frac{\sin 2\alpha }{1+\cos 2\alpha }.\frac{\cos \alpha }{1+\cos \alpha }$

$=\frac{2\sin \alpha \cos \alpha }{2{\cos }^2\alpha }.\frac{\cos \alpha }{1+\cos \alpha }$

$=\frac{\sin \alpha }{1+\cos \alpha }$

$=\frac{2\sin \frac{\alpha }{2}\cos \frac{\alpha }{2}}{2{\cos }^2\frac{\alpha }{2}}$

$=\tan \frac{\alpha }{2}=RHS$