# How do you prove that sectheta/tantheta=sintheta is not an identity by showing a counterexample?

$\tan \frac{\theta}{\sec} \theta = \sin \theta$

#### Explanation:

Let's first simplify the left side and see what it actually equals, then we can come up with a counterexample:

$\sec \frac{\theta}{\tan} \theta = \sin \theta$

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

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

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

$\frac{1}{\sin} \theta \ne \sin \theta$

So we know that in the original equation, the left side of the equation equals the inverse of the right side. So we can write as a counterexample:

$\tan \frac{\theta}{\sec} \theta = \sin \theta$