Simplify the expression by using a double-angle formula.
\[
\cos ^{2} 2 \theta-\sin ^{2} 2 \theta
\]

Answer

Expert–verified

Hide Steps

Answer

Final Answer: The simplified expression is \(\boxed{\cos(4\theta)}\).

Steps

Step 1 ：Given the expression \(\cos ^{2} 2 \theta-\sin ^{2} 2 \theta\).

Step 2 ：Recognize that this is a difference of squares, which can be factored as \((a-b)(a+b)\).

Step 3 ：However, in this case, we can use the double angle formula for cosine, which is \(\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)\).

Step 4 ：Therefore, the given expression can be simplified to \(\cos(4\theta)\).

Step 5 ：Final Answer: The simplified expression is \(\boxed{\cos(4\theta)}\).