How do you simplify f(theta)=-2csc(theta/4)+tan(theta/2)-3cos(theta/4) to trigonometric functions of a unit theta?

Dec 11, 2016

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

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

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

Some identities to substitute in (half-angle and quarter-angle formulas):
$\sin \left(\frac{a}{2}\right) = \pm \sqrt{\frac{1 - \cos a}{2}}$
$\cos \left(\frac{a}{2}\right) = \pm \sqrt{\frac{1 + \cos a}{2}}$
$\sin \left(\frac{\theta}{4}\right) = \sin \left(\frac{\frac{\theta}{2}}{2}\right) = \pm \sqrt{\frac{1 - \pm \sqrt{\frac{1 + \cos \theta}{2}}}{2}}$
$\cos \left(\frac{\theta}{4}\right) = \cos \left(\frac{\frac{\theta}{2}}{2}\right) = \pm \sqrt{\frac{1 + \pm \sqrt{\frac{1 - \cos \theta}{2}}}{2}}$