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

Mar 18, 2017

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

#### Explanation:

$f \left(\theta\right) = - \cot \left(\frac{\theta}{2}\right) + 2 \sec \left(\frac{\theta}{4} + \frac{\pi}{2}\right)$

and as $\sec \left(\frac{\pi}{2} + A\right) = - \csc A$

$f \left(\theta\right) = - \cot \left(\frac{\theta}{2}\right) - 2 \csc \left(\frac{\theta}{4}\right)$

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

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

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

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

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

= $- \frac{\sqrt{1 + \cos \theta} + 2 \sqrt{2}}{\sqrt{1 - \cos \theta}}$

multiplying numerator and denominator by $\sqrt{1 + \cos \theta}$, we get

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

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