# How do you simplify f(theta)=2cot(theta/2)-3sin(theta/4+pi/2) to trigonometric functions of a unit theta?

Aug 9, 2018

$\pm 2 \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}$
$- \frac{3}{\sqrt{2}} \sqrt{1 \pm \frac{1}{\sqrt{2}} \sqrt{1 + \cos \theta}}$. See explanation on +_ dilemma.

#### Explanation:

As an advocate of $\theta \ge 0$, like $r \ge 0$, I assume that $\theta$ is non-

negative.
$2 \cot \left(\frac{\theta}{2}\right) - 3 \sin \left(\frac{\pi}{2} + \frac{\theta}{4}\right)$

$= 2 \cos \frac{\frac{\theta}{2}}{\sin} \left(\frac{\theta}{2}\right) - 3 \cos \left(\frac{\theta}{4}\right)$

$= \pm 2 \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}$

$- \frac{3}{\sqrt{2}} \sqrt{1 + \cos \left(\frac{\theta}{2}\right)}$

$= \pm 2 \sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}$

$- \frac{3}{\sqrt{2}} \sqrt{1 \pm \frac{1}{\sqrt{2}} \sqrt{1 + \cos \theta}}$, choosing $+$

prefix for the first term, when $\theta \in {Q}_{4}$ and, for the same

reason, $-$ in the 3rd term, under the radical sign.