How do you prove ## cos ^2theta - sin ^2 theta = 1 - 2sin^2 theta ##?

The Pythagorean identity states that ##sin^2theta+cos^2theta=1##.

We can rearrange the identity to see that ##cos^2theta=1-sin^2theta##.

If we know that ##cos^2theta=1-sin^2theta##, we can replace ##cos^2theta## with ##1-sin^2theta## in the expression ##cos^2theta-sin^2theta##.

##cos^2theta-sin^2theta=(1color(blue)(-sin^2theta))color(blue)(-sin^2theta##

Combine like terms to see that:

##cos^2theta-sin^2theta=1-2sin^2theta##