Expand \(\cos^2x\) using double-angle formulas.

Step 1 ：First, recall the double-angle formula for cosine: \(\cos2x = 1 - 2\sin^2x\) or \(\cos2x = 2\cos^2x - 1\).

Step 2 ：We need to express \(\cos^2x\) in terms of \(\cos 2x\), so we rearrange the second double-angle formula to solve for \(\cos^2x\): \(\cos^2x = \frac{\cos2x + 1}{2}\)