To prove the identity \(\cos 4\theta - 4\cos 2\theta \equiv 8\sin^4 \theta - 3\), we start by using trigonometric identities.

1. Express \(\cos 4\theta\) using the double angle formula:

\(\cos 4\theta = 2\cos^2 2\theta - 1\)

2. Express \(\cos 2\theta\) using the double angle formula:

\(\cos 2\theta = 1 - 2\sin^2 \theta\)

3. Substitute \(\cos 2\theta\) into \(\cos 4\theta\):

\(\cos 4\theta = 2(1 - 2\sin^2 \theta)^2 - 1\)

\(= 2(1 - 4\sin^2 \theta + 4\sin^4 \theta) - 1\)

\(= 2 - 8\sin^2 \theta + 8\sin^4 \theta - 1\)

\(= 8\sin^4 \theta - 8\sin^2 \theta + 1\)

4. Substitute into the left-hand side of the identity:

\(\cos 4\theta - 4\cos 2\theta = (8\sin^4 \theta - 8\sin^2 \theta + 1) - 4(1 - 2\sin^2 \theta)\)

\(= 8\sin^4 \theta - 8\sin^2 \theta + 1 - 4 + 8\sin^2 \theta\)

\(= 8\sin^4 \theta - 3\)

Thus, the identity is proven.