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

Feb 11, 2018

$f \left(\theta\right) = \frac{1}{2} \left(\cot \left(\frac{\theta}{4}\right) - \tan \left(\frac{\theta}{4}\right)\right) + 3 \cos \left(\frac{\theta}{4}\right)$

#### Explanation:

Given:
$f \left(\theta\right) = \cot \left(\frac{\theta}{2}\right) + 3 \sin \left(\frac{\theta}{4} + \frac{\pi}{2}\right)$
$\cot \left(\frac{\theta}{2}\right) = \cos \frac{\frac{\theta}{2}}{\sin} \left(\frac{\theta}{2}\right)$
$\sin \left(\frac{\theta}{4} + \frac{\pi}{2}\right) = \sin \left(\frac{\pi}{2} + \frac{\theta}{4}\right)$
$= \cos \left(\frac{\theta}{4}\right)$
$3 \sin \left(\frac{\theta}{4} + \frac{\pi}{2}\right) = 3 \cos \left(\frac{\theta}{4}\right)$
Thus,
$\cot \left(\frac{\theta}{2}\right) + 3 \sin \left(\frac{\theta}{4} + \frac{\pi}{2}\right) = \cos \frac{\frac{\theta}{2}}{\sin} \left(\frac{\theta}{2}\right) + 3 \cos \left(\frac{\theta}{4}\right)$
$\cos \left(\frac{\theta}{2}\right) = \cos 2 \left(\frac{\theta}{4}\right)$

$\cos \left(\frac{\theta}{2}\right) = {\cos}^{2} \left(\frac{\theta}{4}\right) - {\sin}^{2} \left(\frac{\theta}{4}\right)$
$\sin \left(\frac{\theta}{2}\right) = \sin 2 \left(\frac{\theta}{4}\right)$
$\sin \left(\frac{\theta}{2}\right) = 2 \sin \left(\frac{\theta}{4}\right) \cos \left(\frac{\theta}{4}\right)$
Thus,
cot(theta/2)=(cos^2(theta/4)-sin^2(theta/4))/(2sin(theta/4)cos(theta/4)
$\cot \left(\frac{\theta}{2}\right) = \frac{1}{2} \left(\cot \left(\frac{\theta}{4}\right) - \tan \left(\frac{\theta}{4}\right)\right)$

Now,

$\cot \left(\frac{\theta}{2}\right) + 3 \sin \left(\frac{\theta}{4} + \frac{\pi}{2}\right) = \frac{1}{2} \left(\cot \left(\frac{\theta}{4}\right) - \tan \left(\frac{\theta}{4}\right)\right) + 3 \cos \left(\frac{\theta}{4}\right)$
$f \left(\theta\right) = \frac{1}{2} \left(\cot \left(\frac{\theta}{4}\right) - \tan \left(\frac{\theta}{4}\right)\right) + 3 \cos \left(\frac{\theta}{4}\right)$