Simplify each expression. See Example 4.
cos² π/8 - 1/2

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Recognize that the expression is \(\cos^{2} \frac{\pi}{8} - \frac{1}{2}\), where \(\cos^{2} \theta\) means \((\cos \theta)^2\).

Recall the double-angle identity for cosine: \(\cos 2\theta = 2\cos^{2} \theta - 1\). This can be rearranged to express \(\cos^{2} \theta\) as \(\cos^{2} \theta = \frac{1 + \cos 2\theta}{2}\).

Apply this identity to \(\cos^{2} \frac{\pi}{8}\) by substituting \(\theta = \frac{\pi}{8}\), so \(\cos^{2} \frac{\pi}{8} = \frac{1 + \cos \frac{\pi}{4}}{2}\).

Substitute this back into the original expression: \(\cos^{2} \frac{\pi}{8} - \frac{1}{2} = \frac{1 + \cos \frac{\pi}{4}}{2} - \frac{1}{2}\).

Simplify the expression by combining the fractions: \(\frac{1 + \cos \frac{\pi}{4} - 1}{2} = \frac{\cos \frac{\pi}{4}}{2}\). This is the simplified form in terms of cosine.

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