Answer
$$\cos^22x-\sin^22x=\cos4x$$
Work Step by Step
$$X=\cos^22x-\sin^22x$$
Recall the Double-Angle Identity for cosine, which states
$$\cos^2A-\sin^2A=\cos2A$$
Therefore, here we can absolutely apply the identity right away with $A=2x$, meaning that
$$\cos^22x-\sin^22x=\cos(2\times2x)$$
$$\cos^22x-\sin^22x=\cos4x$$
Therefore,
$$X=\cos4x$$
In conclusion, the result is
$$\cos^22x-\sin^22x=\cos4x$$
