Use a compound angle formula for cosine to show that cos2x = cos^2x - sin^2x

To prove the compound angle formula for cosine, we'll start by using the double angle formula for cosine.

The double angle formula for cosine states that cos(2x) = cos^2(x) - sin^2(x).

To derive this formula, we'll begin with the sum formula for cosine:

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Now, let's substitute A = x and B = x into the above formula:

cos(2x) = cos(x + x) = cos(x)cos(x) - sin(x)sin(x) = cos^2(x) - sin^2(x)

So, we have shown that cos(2x) = cos^2(x) - sin^2(x), which is the compound angle formula for cosine.

Therefore, cos(2x) = cos^2(x) - sin^2(x).