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).