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

Feb 8, 2016

$1 - {\cos}^{2} \theta - \frac{1}{\cos} \theta + \frac{1}{1 - {\cos}^{2} \theta}$

#### Explanation:

You should use the following identities:

[1] $\text{ "sin^2 theta + cos^2 theta = 1 " "<=>" } {\sin}^{2} \theta = 1 - {\cos}^{2} \theta$

[2] $\text{ } \sec \theta = \frac{1}{\cos} \theta$

[3] $\text{ } \csc \theta = \frac{1}{\sin} \theta$

Thus, your expression can be transformed as follows:

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

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

.... use ${\sin}^{2} \theta = 1 - {\cos}^{2} \theta$ once again...

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