Question

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

Answer 1

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`