# How do you express tan theta - cot theta +sintheta  in terms of cos theta ?

Jan 9, 2016

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

#### Explanation:

$\tan \left(\theta\right) - \cot \left(\theta\right) + \sin \left(\theta\right)$
We have to write in terms of $\cos \left(\theta\right)$

$\textcolor{b l u e}{\text{Let us start by using the identity}}$
$\tan \left(\theta\right) = \sin \frac{\theta}{\cos} \left(\theta\right)$ and $\cot \left(\theta\right) = \cos \frac{\theta}{\sin} \left(\theta\right)$

We get

$\tan \left(\theta\right) - \cot \left(\theta\right) + \sin \left(\theta\right)$
$= \sin \frac{\theta}{\cos} \left(\theta\right) - \cos \frac{\theta}{\sin} \left(\theta\right) + \sin \left(\theta\right)$

$\textcolor{b l u e}{\text{In order to simplify we need to use Least Common Denominator for all the fractions}}$

$= \frac{\sin \left(\theta\right) \sin \left(\theta\right)}{\cos \left(\theta\right) \sin \left(\theta\right)} - \frac{\cos \left(\theta\right) \cos \left(\theta\right)}{\cos \left(\theta\right) \sin \left(\theta\right)} + \frac{\sin \left(\theta\right) \cos \left(\theta\right) \sin \left(\theta\right)}{\cos \left(\theta\right) \sin \left(\theta\right)}$

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

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

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

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

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