# What is cottheta-costheta in terms of sintheta?

Dec 6, 2015

$\frac{{\left(1 - \sin \left(x\right)\right)}^{\frac{3}{2}} \sqrt{1 + \sin \left(x\right)}}{\sin \left(x\right)}$

#### Explanation:

We first have to put everything at the same denominator.

$\cos \frac{x}{\sin} \left(x\right) - \cos \left(x\right) = \frac{\cos \left(x\right) - \sin \left(x\right) . \cos \left(x\right)}{\sin \left(x\right)} = \frac{\left(\cos \left(x\right)\right) \left(1 - \sin \left(x\right)\right)}{\sin \left(x\right)}$

We know that :
$\cos \left(x\right) = \sqrt{1 - {\sin}^{2} \left(x\right)} = \sqrt{1 - \sin \left(x\right)} \sqrt{1 + \sin \left(x\right)} .$

Therefor,
$\cot \left(x\right) - \cos \left(x\right) = \frac{{\left(1 - \sin \left(x\right)\right)}^{\frac{3}{2}} \sqrt{1 + \sin \left(x\right)}}{\sin \left(x\right)}$