Answer
$$\cos2\theta=\frac{2-\sec^2\theta}{\sec^2\theta}$$
Both sides are equal, so the equation is an identity.
Work Step by Step
$$\cos2\theta=\frac{2-\sec^2\theta}{\sec^2\theta}$$
We start from the right side and try to make it equal to the left side.
$$X=\frac{2-\sec^2\theta}{\sec^2\theta}$$
- Reciprocal Identity: $\sec\theta=\frac{1}{\cos\theta}$, meaning $\sec^2\theta=\frac{1}{\cos^2\theta}$
Then, $$X=\frac{2-\frac{1}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}$$
$$X=\frac{\frac{2\cos^2\theta-1}{\cos^2\theta}}{\frac{1}{\cos^2\theta}}$$
$$X=\frac{2\cos^2\theta-1}{1}$$
$$X=2\cos^2\theta-1$$
- Double-Angle Identity: $2\cos^2\theta-1=\cos2\theta$
Thus, $$X=\cos2\theta$$
Therefore, $$\cos2\theta=\frac{2-\sec^2\theta}{\sec^2\theta}$$
Since both sides are equal, the equation is an identity.
