Cos^theta - sin^2 theta = 1-2 sin^2 theta

Yes indeed. That's a true statement.

Let's just work on the left side, and leave the right side alone.

The left side is cos² - sin².

Do you remember the trig identity that says cos² + sin² = 1?

Take that and subtract sin² from each side of it: cos² = 1 - sin²

Now substitute that into the left side of the problem:

Left side: cos² - sin²

Substitute: (1 - sin²) - sin²

Remove parentheses: 1 - sin² - sin²

Combine like terms: 1 - 2 sin²

Now compare that to the right side of the problem.