# How do you simplify (cot θ + tan θ) sec θ?

Jun 23, 2016

Recall that $\tan \theta = \sin \frac{\theta}{\cos} \theta$.

#### Explanation:

Since $\cot \theta = \frac{1}{\tan} \theta$, cottheta = 1/(sintheta/costheta)= costheta/sintheta.

Also, $\sec \theta = \frac{1}{\cos} \theta$.

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

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

Recall the Pythagorean Identity ${\sin}^{2} \theta + {\cos}^{2} \theta = 1$:

$= \frac{1}{\cos \theta \sin \theta \cos \theta}$

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

Use the rearranged form of the Pythagorean identity presented above.

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

=1/(sintheta - sin^3theta#

We could have also finished with ${\sec}^{2} \theta \csc \theta$, because $\frac{1}{\sin} \theta = \csc \theta$ and $\frac{1}{\cos} \theta = \sec \theta$

Hopefully this helps!