# What is cos(theta/2)-2cot(theta/2) in terms of trigonometric functions of a unit theta?

Sep 18, 2016

{sintheta-2sqrt(2(1+costheta))}/sqrt(2(1-costheta).

#### Explanation:

$\cos \left(\frac{\theta}{2}\right) - 2 \cot \left(\frac{\theta}{2}\right) = \cos \left(\frac{\theta}{2}\right) - 2 \cdot \cos \frac{\frac{\theta}{2}}{\sin} \left(\frac{\theta}{2}\right)$

$= \frac{\cos \left(\frac{\theta}{2}\right) \sin \left(\frac{\theta}{2}\right) - 2 \cos \left(\frac{\theta}{2}\right)}{\sin} \left(\frac{\theta}{2}\right)$.

Multiplying by $2$ in $N r . \mathmr{and} D r .$, we get,

the Exp.$= \frac{2 \cos \left(\frac{\theta}{2}\right) \sin \left(\frac{\theta}{2}\right) - 4 \cos \left(\frac{\theta}{2}\right)}{2 \sin \left(\frac{\theta}{2}\right)}$.

Here, we use the Identities :

 (i) :2sinxcosx=sin2x; (ii) : 1-cos2x=2sin^2x;
$\left(i i i\right) : 1 + \cos 2 x = 2 {\cos}^{2} x$.

Hence,

"The Exp.="{sintheta-4sqrt((1+costheta)/2)}/(2sqrt((1-costheta)/2)

={sintheta-2sqrt(2(1+costheta))}/sqrt(2(1-costheta).

Enjoy Maths.!