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

Aug 12, 2018

+-sqrt(( 1 - cos theta )/( 1 + cos theta )
+ sqrt2 sqrt(1/( 1 +- 1/sqrt2 sqrt( 1 + cos theta ))
$- \frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}} \sqrt{1 + \cos \theta}}$

#### Explanation:

scalar r = sqrt ( x^2 + y^2 ) >= 0.

Of course, sooner or later $\theta$, used in measuring clockwise or

anticlockwise time-oriented Natural rotations and revolutions, and

serving as a $\pi$-oriented location-marker, in the marker couple

$\left(r , \theta\right)$, would be declared as non-negative.

And now, I assume that $\theta \ge 0$.

For this problem, if $0 \le \theta \in {Q}_{4} , \frac{\theta}{2} \in Q - 2$.

Accordingly, $\cos \left(\frac{\theta}{2}\right) \mathmr{and} \tan \left(\frac{\theta}{2}\right) \le 0.$

f ( theta ) = +-sqrt(( 1 - cos theta )/( 1 + cos theta )

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

= +-sqrt(( 1 - cos theta )/( 1 + cos theta )

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

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

Sign Disambiguation:

If $\theta \notin {Q}_{4}$, choose + for 1st and 2nd terms

and $-$ for the 3rd. Otherwise. they are opposites.