# How do you simplify f(theta)=-2csc(theta/4)-cot(theta/2)+2sin(theta/4) to trigonometric functions of a unit theta?

Jul 23, 2016

$= \frac{- 2 \left(1 + \sqrt{\frac{1}{2} \left(1 + \cos \theta\right)}\right)}{\sqrt{\frac{1}{2} \left(1 - \sqrt{\frac{1}{2} \left(1 + \cos \theta\right)}\right)}} - \frac{1 + \cos \theta}{\sin} \theta$

#### Explanation:

$f \left(\theta\right) = - 2 \csc \left(\frac{\theta}{4}\right) - \cot \left(\frac{\theta}{2}\right) + 2 \sin \left(\frac{\theta}{4}\right)$

$= 2 \sin \left(\frac{\theta}{4}\right) - \frac{2}{\sin} \left(\frac{\theta}{4}\right) - \cot \left(\frac{\theta}{2}\right)$

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

$= \frac{- 2 \left(1 - {\sin}^{2} \left(\frac{\theta}{4}\right)\right)}{\sin} \left(\frac{\theta}{4}\right) - \frac{1 + \cos \theta}{\sin} \theta$

$= \frac{- 2 {\cos}^{2} \left(\frac{\theta}{4}\right)}{\sin} \left(\frac{\theta}{4}\right) - \frac{1 + \cos \theta}{\sin} \theta$

$= \frac{- 2 \left(1 + \cos \left(\frac{\theta}{2}\right)\right)}{\sqrt{\frac{1}{2} \left(1 - \cos \left(\frac{\theta}{2}\right)\right)}} - \frac{1 + \cos \theta}{\sin} \theta$

$= \frac{- 2 \left(1 + \sqrt{\frac{1}{2} \left(1 + \cos \theta\right)}\right)}{\sqrt{\frac{1}{2} \left(1 - \cos \left(\frac{\theta}{2}\right)\right)}} - \frac{1 + \cos \theta}{\sin} \theta$

$= \frac{- 2 \left(1 + \sqrt{\frac{1}{2} \left(1 + \cos \theta\right)}\right)}{\sqrt{\frac{1}{2} \left(1 - \sqrt{\frac{1}{2} \left(1 + \cos \theta\right)}\right)}} - \frac{1 + \cos \theta}{\sin} \theta$