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.