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

Mar 1, 2017

$f \left(\theta\right) = \cot \left(\frac{\theta}{4}\right) - \tan \left(\frac{\theta}{2} + \frac{\pi}{2}\right) = \sqrt{\frac{2}{1 - \cos \theta}}$

#### Explanation:

As $\tan \left({90}^{\circ} + A\right) = - \cot A$, $\tan \left(\frac{\theta}{2} + \frac{\pi}{2}\right) = - \cot \left(\frac{\theta}{2}\right)$

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

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

Further cot(A/2)=cos(A/2)/sin(A/2)=(2cos^2(A/2))/(2sin(A/2)cos(A/2)

= $\frac{1 + \cos A}{\sin} A = \csc A + \cot A$

Also $\sin \left(\frac{A}{2}\right) = \sqrt{\frac{2 {\sin}^{2} \left(\frac{A}{2}\right)}{2}} = \sqrt{\frac{1 - \left(1 - 2 {\sin}^{2} \left(\frac{A}{2}\right)\right)}{2}}$

= $\sqrt{\frac{1 - \cos A}{2}}$ .......................(A)

Hence, $\cot \left(\frac{\theta}{4}\right) - \cot \left(\frac{\theta}{2}\right)$

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

= $\csc \left(\frac{\theta}{2}\right)$

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

= $\sqrt{\frac{2}{1 - \cos \theta}}$ (using (A))