3. Verify the identity. Your answer for the identity will be your verification steps/process. \[ \cos ^{8} x-\sin ^{8} x=\cos 2 x\left(\cos ^{4} x+\sin ^{4} x\right) \]

Step 1 ：The left side of the equation is a difference of squares, which can be factored into \((\cos^4x + \sin^4x)(\cos^4x - \sin^4x)\).

Step 2 ：The right side of the equation is \(\cos 2x(\cos^4x + \sin^4x)\).

Step 3 ：We know that \(\cos^2x + \sin^2x = 1\), so \(\cos^4x + \sin^4x\) is a common factor on both sides.

Step 4 ：We can simplify \(\cos^4x - \sin^4x\) to \(\cos 2x\) using the double angle identity \(\cos 2x = \cos^2x - \sin^2x\).

Step 5 ：Therefore, the left side of the equation can be simplified to the right side of the equation, verifying the identity.

Step 6 ：Final Answer: The identity \(\cos ^{8} x-\sin ^{8} x=\cos 2 x\left(\cos ^{4} x+\sin ^{4} x\right)\) is \(\boxed{True}\).