# How do you express cos theta - cos^2 theta + cot^2 theta  in terms of sin theta ?

Jan 15, 2016

$\frac{2 {\sin}^{4} \theta - 4 {\sin}^{2} \theta + \sin \theta \sin 2 \theta + 2}{2 {\sin}^{2} \theta}$

#### Explanation:

Write in terms of $\sin \theta$ and $\cos \theta$.

$= \cos \theta - {\cos}^{2} \theta + {\cos}^{2} \frac{\theta}{\sin} ^ 2 \theta$

Find a common denominator.

$= \frac{\cos \theta {\sin}^{2} \theta}{\sin} ^ 2 \theta - \frac{{\cos}^{2} \theta {\sin}^{2} \theta}{\sin} ^ 2 \theta + {\cos}^{2} \frac{\theta}{\sin} ^ 2 \theta$

Combine.

$= \frac{\cos \theta {\sin}^{2} \theta - {\cos}^{2} \theta {\sin}^{2} \theta + {\cos}^{2} \theta}{\sin} ^ 2 \theta$

The following simplification may seem unecessary, but is actually relevant. Its purpose will become clear in the following step.

$= \frac{\sin \theta \left(\textcolor{b l u e}{\cos \theta \sin \theta}\right) - \textcolor{g r e e n}{{\cos}^{2} \theta} {\sin}^{2} \theta + \textcolor{g r e e n}{{\cos}^{2} \theta}}{\sin} ^ 2 \theta$

Use the following identities:

• color(green)(cos^2theta=1-sin^2theta
• 2costhetasintheta=sin2theta=>color(blue)(costhetasintheta=(sin2theta)/2

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

$= \frac{\frac{\sin \theta \sin 2 \theta}{2} - {\sin}^{2} \theta + {\sin}^{4} \theta + 1 - {\sin}^{2} \theta}{\sin} ^ 2 \theta$

$= \frac{\frac{\sin \theta \sin 2 \theta}{2} - 2 {\sin}^{2} \theta + {\sin}^{4} \theta + 1}{\sin} ^ 2 \theta$

$= \frac{2 {\sin}^{4} \theta - 4 {\sin}^{2} \theta + \sin \theta \sin 2 \theta + 2}{2 {\sin}^{2} \theta}$