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

##### 1 Answer
Jul 24, 2018

color(indigo)(f(theta) = (2 * pm sqrt((1 + cos theta) / (1 - cos theta))) - pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta))))) - pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))

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

color(brown)(cot (theta/2) = 1 / tan (theta/2) = pm sqrt((1 + cos theta) / (1 - cos theta))

$\sin \left(\frac{\theta}{4}\right) = \pm \sqrt{\frac{1}{2} \left(1 - \sin \left(\frac{\theta}{2}\right)\right)}$

color(brown)(csc (theta/4) = 1 / sin (theta / 4) = pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta)))

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

color(brown)(cos (theta / 4) = pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))

color(indigo)(f(theta) = (2 * pm sqrt((1 + cos theta) / (1 - cos theta))) - pm 1 / (sqrt(1/2 (1 - pm sqrt(1/2 (1 - cos theta))))) - pm sqrt(1/2 (1 pm sqrt(1/2 (1 + cos theta)))