# How do you express f(theta)=cos(theta/4)+csc(theta/2)+sin^2(theta/2) in terms of trigonometric functions of a whole theta?

Nov 28, 2016

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

#### Explanation:

We are going to use,

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

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

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

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

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

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

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

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

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

And finally,

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