Question2 Rewrite $\cos 4 x$ in terms of $\cos x$ and powers of $\cos x$. ( 3 points) (Note: The answer cannot have multiple angles of cosines)

Studdy Solution
Expand and simplify the expression:
First, expand $(2\cos^2 x - 1)^2$:
$(2\cos^2 x - 1)^2 = 4\cos^4 x - 4\cos^2 x + 1$
Substitute back into the expression for $\cos 4x$:
$\cos 4x = 2(4\cos^4 x - 4\cos^2 x + 1) - 1$
Simplify:
$\cos 4x = 8\cos^4 x - 8\cos^2 x + 2 - 1$
$\cos 4x = 8\cos^4 x - 8\cos^2 x + 1$
The expression for $\cos 4x$ in terms of $\cos x$ is:
$\boxed{8\cos^4 x - 8\cos^2 x + 1}$