Answer
$$\tan2\theta=\frac{-2\tan\theta}{\sec^2\theta-2}$$
The proof that 2 sides are equal to each other is below. As they are equal, the equation is an identity.
Work Step by Step
$$\tan2\theta=\frac{-2\tan\theta}{\sec^2\theta-2}$$
We start from where it is more complicated, which is the right side in this exercise.
$$X=\frac{-2\tan\theta}{\sec^2\theta-2}$$
- We get from Pythagorean Identities that $\sec^2\theta=\tan^2\theta+1$. Thus, we can replace $\sec^2\theta$ in $X$ with $\tan^2\theta+1$.
$$X=\frac{-2\tan\theta}{\tan^2\theta+1-2}$$
$$X=\frac{-2\tan\theta}{\tan^2\theta-1}$$
$$X=\frac{2\tan\theta}{-(\tan^2\theta-1)}$$
$$X=\frac{2\tan\theta}{1-\tan^2\theta}$$
- From Double-Angel Identity: $\tan2\theta=\frac{2\tan\theta}{1-\tan^2\theta}$
$$X=\tan2\theta$$
Therefore, $$\tan2\theta=\frac{-2\tan\theta}{\sec^2\theta-2}$$
As we have proved 2 sides are equal to each other, we conclude that the equation is an identity.
