# What is sectheta+cos^2theta -costheta in terms of sintheta?

Jan 22, 2016

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

#### Explanation:

$\sec \left(\theta\right) + {\cos}^{2} \left(\theta\right) - \cos \left(\theta\right)$

To simplify in terms of $\sin \left(\theta\right)$ let us write $\sec \left(\theta\right)$ as $\frac{1}{\cos} \left(\theta\right)$

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

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

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

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

=(sin^2(theta)+sqrt(1-sin^2(theta))(1-sin^2(theta)))/sqrt(1-sin^2(theta)

If you need it can be simplified further as

=sin^2(theta)/sqrt(1-sin^2(theta)) + (sqrt(1-sin^2(theta))(1-sin^2(theta)))/sqrt(1-sin^2(theta)

=sin^2(theta)/sqrt(1-sin^2(theta)) + (cancel(sqrt(1-sin^2(theta)))(1-sin^2(theta)))/cancel(sqrt(1-sin^2(theta))

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