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

Oct 21, 2017

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

#### Explanation:

Let $\theta$ = x

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

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

$\csc \left(\frac{x}{2}\right)$ --> $\sec \left(x\right) = 1 - 2 {\csc}^{2} \left(\frac{x}{2}\right)$ --> $\sqrt{\frac{2 + \cos x}{2 \cos x}}$

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

I could only come up to this, i'm pretty sure it can be simplified further, however, i've done the bulk of it.