Question

Using the double angle of half angle formula, how do you simplify `cos^2 5theta- sin^2 5theta`?

Answer 1

There is another simple way to simplify this.
`cos^2 5x - sin^2 5x = (cos 5x - sin 5x)(cos 5x + sin 5x)`
Use the identities:
`cos a - sin a = -(sqrt2) *(sin (a - Pi/4))`

`cos a + sin a = (sqrt2)* (sin (a + Pi/4))`

So this becomes:
`-2 * sin (5x - Pi/4) * sin (5x + Pi/4)`.

Since `sin a * sin b = 1/2 (cos(a-b)-cos(a+b))`, this equation can be rephrased as (removing the parentheses inside the cosine):

`-(cos(5x - Pi/4-5x-Pi/4)-cos(5x - Pi/4+5x + Pi/4))`

This simplifies to:

`-(cos(-pi/2)-cos(10x))`

The cosine of `-pi/2` is 0, so this becomes:

`-(-cos(10x))`

`cos(10x)`

Unless my math is wrong, this is the simplified answer.