# How do you express sin theta - csc theta + sec theta  in terms of cos theta ?

Jan 12, 2016

=(1-cos^3(theta))/(cos(theta)sqrt(1-cos^2(theta))

#### Explanation:

$\sin \left(\theta\right) - \csc \left(\theta\right) + \sec \left(\theta\right)$

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

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

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

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

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

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

=(-cos^3(theta)+1)/(cos(theta)sqrt(1-cos^2(theta))

=(1-cos^3(theta))/(cos(theta)sqrt(1-cos^2(theta))