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

Sep 1, 2017

$\cos \left(\frac{\theta}{2}\right) - \tan \left(\frac{\theta}{2}\right) + \sin \left(\frac{\theta}{2}\right)$

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

#### Explanation:

Recall $\cos 2 A = 2 {\cos}^{2} A - 1 = 1 - 2 {\sin}^{2} A$

if $A = \frac{\theta}{2}$. we have

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

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

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

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

and dividing (2) by (1), we get

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

We use only positive sign for $\cos \left(\frac{\theta}{2}\right) , \tan \left(\frac{\theta}{2}\right) , \sin \left(\frac{\theta}{2}\right)$

for working out the desired result, although multiple values of $f \left(\theta\right)$ are possible and we get

$\cos \left(\frac{\theta}{2}\right) - \tan \left(\frac{\theta}{2}\right) + \sin \left(\frac{\theta}{2}\right)$

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