# How do you express 4 cos^2 theta - sec^2 theta + 2 cot theta  in terms of sin theta ?

Feb 19, 2016

$4 - 4 {\sin}^{2} \theta - \frac{1}{1 - {\sin}^{2} \theta} + \frac{2 \sqrt{1 - {\sin}^{2} \theta}}{\sin} \theta$

#### Explanation:

We will use the following identities:

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

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

[3] $\text{ } \cot \theta = \cos \frac{\theta}{\sin} \theta$

Thus, we can express the term as follows:

$4 {\cos}^{2} \theta - {\sec}^{2} \theta + 2 \cot \theta$

$= 4 \left(1 - {\sin}^{2} \theta\right) - \frac{1}{\cos} ^ 2 \theta + \frac{2 \cos \theta}{\sin} \theta$

... apply [1] once again...

$= 4 - 4 {\sin}^{2} \theta - \frac{1}{1 - {\sin}^{2} \theta} + \frac{2 \cos \theta}{\sin} \theta$

... now, there is only one $\cos \theta$ expression left which we can express as $\sqrt{1 - {\sin}^{2} \theta}$:

$= 4 - 4 {\sin}^{2} \theta - \frac{1}{1 - {\sin}^{2} \theta} + \frac{2 \sqrt{1 - {\sin}^{2} \theta}}{\sin} \theta$

Hope that this helped!