Question

How do you use the double angle or half angle formulas to simplify `cos^2 4x - sin^2 4x`?

Answer 1

`cos(8x)`.

Explanation:

We know that `cos(2y)=cos^2(y)-sin^2(y)`. So, in your case, the role of `y` is being played by `4x`.

Trying to convert the formula in words, you can write `cos^2(y)-sin^2(y)=cos(2y)` as

"the difference between the squares of the cosine and the sine of an angle is the cosine of twice that angle".

And what do you have? `cos^2(4x)-sin^2(4x)` is exactly the difference between the squares of the cosine and the sine of an angle, namely of `4x`. This means that `cos^2(4x)-sin^2(4x)` is the cosine of twice the angle, namely `2*4x=8x`.