# What is cottheta-sectheta+costheta in terms of sintheta?

Jan 15, 2016

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

#### Explanation:

Express everything in terms of $\sin \theta$ and $\cos \theta$.

$= \cos \frac{\theta}{\sin} \theta - \frac{1}{\cos} \theta + \cos \theta$

Get a common denominator.

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

Combine.

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

Here, use the identities:

• ${\cos}^{2} \theta = 1 - {\sin}^{2} \theta$
• $2 \cos \theta \sin \theta = \sin 2 \theta \implies \cos \theta \sin \theta = \frac{\sin 2 \theta}{2}$

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

$= \frac{1 - {\sin}^{2} \theta - \sin \theta + \sin - {\sin}^{3} \theta}{\frac{\sin 2 \theta}{2}}$

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