Answer
$$\cos2x=\frac{2-\sec^2x}{\sec^2x}$$
The equation is verified to be an identity as below.
Work Step by Step
$$\cos2x=\frac{2-\sec^2x}{\sec^2x}$$
In this exercise, we would tackle the right side first.
$$X=\frac{2-\sec^2x}{\sec^2x}$$
- For $\sec x$, we rewrite according to the following reciprocal identity:
$$\sec x=\frac{1}{\cos x}$$
Apply to $X$:
$$X=\frac{2-\frac{1}{\cos^2x}}{\frac{1}{\cos^2x}}$$
$$X=\frac{\frac{2\cos^2x-1}{\cos^2x}}{\frac{1}{\cos^2x}}$$
$$X=\frac{2\cos^2x-1}{1}$$
$$X=2\cos^2x-1$$
- Now recall that $2\cos^2x-1=\cos2x$. Therefore,
$$X=\cos2x$$
That means $$\cos2x=\frac{2-\sec^2x}{\sec^2x}$$
As 2 sides are equal, the equation is an identity.
