# How do you simplify f(theta)=csc2theta-sec2theta-3cot2theta to trigonometric functions of a unit theta?

Oct 14, 2017

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

#### Explanation:

Break the function down by term:
$\csc \left(2 \theta\right) = \frac{1}{\sin \left(2 \theta\right)} = \frac{1}{2 \sin \theta \cos \theta}$

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

$\cot \left(2 \theta\right) = \frac{1}{\tan} \left(2 \theta\right) = \frac{1 - {\tan}^{2} \theta}{2 \tan \theta}$

Put this hell together:

$f \left(\theta\right) = \frac{1}{\sin \left(2 \theta\right)} = \frac{1}{2 \sin \theta \cos \theta} - \frac{1}{2 {\cos}^{2} \theta - 1} - \frac{3 - 3 {\tan}^{2} \theta}{2 \tan \theta}$