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

Aug 8, 2018

#### Explanation:

It is my rule that, on par with $r \ge 0 , \theta \ge 0$

In other words, for Australians, the Earth spins clockwise and, for

Mexicans, it is anticlockwise.

So, both can keep the spin $\theta \ge 0$. This is universal relativity,

with respect to the advancing real time. And so,

if $\theta \in {Q}_{4} , \frac{\theta}{2} \in {Q}_{2}$ and

if $\theta \in {Q}_{3} , \frac{\theta}{2} \in {Q}_{1}$.

For any $\theta , \frac{\theta}{4} \in {Q}_{1}$. Now,

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

$- 4 \frac{\sqrt{2}}{1 + \cos \left(\frac{\theta}{2}\right)} + \frac{1}{\sqrt{2}} \sqrt{1 - \cos \left(\frac{\theta}{2}\right)}$

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

- 4sqrt2 /sqrt (( 1 + sqrt(1/ 2( 1+ cos theta )))

+ 1/sqrt2 sqrt( 1 - sqrt( 1/2( 1 + cos theta) ).

Choose the prefix sign $-$, for the 1st term,

when $0 \le \theta \in {Q}_{4}$.

Example: $\theta = {330}^{o}$, $\frac{\theta}{2} = {165}^{o} \in {Q}_{2}$,

wherein $\tan \left(\frac{\theta}{2}\right) = \tan {165}^{o} = - 0.2679 \ldots < 0$