Question

Prove that $\tan \frac{x}{2}=\text{cosec}x-cotx$

Answer 1

$R.H.S.=cosecx-cotx$

$=\frac{1}{sinx}-\frac{cosx}{sinx}$

$=\frac{1-cosx}{sinx}$

Here,we use the following Identities:

$1-cosx=2{sin}^2\left(\frac{x}{2}\right)$ and $sinx=2sin\left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right).Hence,$

$R.H.S.=\frac{2{sin}^2\left(\frac{x}{2}\right)}{2sin\left(\frac{x}{2}\right)cos\left(\frac{x}{2}\right)}$

$=\frac{sin\left(\frac{x}{2}\right)}{cos\left(\frac{x}{2}\right)}$

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

$=L.H.S.$