Question

Solve:
$\frac{x\sin x}{1+\cos x}$

Answer 1

$\frac{x\sin x}{1+\cos x}=y$ (Let)

Multiply Numerator & Denominator by $1-\cos x$.

$\Rightarrow y=\frac{x\sin (1-\cos x)}{(1-{\cos }^{2}x)}[\because (a+b\left)\right(a-b)=a^2-b^2]$

$\Rightarrow y=\frac{x\sin (1-\cos x)}{{\sin }^{2}x}[\because {\sin }^{2}x+{\cos }^{2}x=1]$

Also $1-\cos x=2{\sin }^2\left(\frac{x}{2}\right)\sin x=2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)$

$\Rightarrow y=\frac{x2{\sin }^2\left(\frac{x}{2}\right)}{2\sin \left(\frac{x}{2}\right)\cos \left(\frac{x}{2}\right)}$

$\Rightarrow y=x\tan \left(\frac{x}{2}\right)$