Find the exact value of each expression. (Do not use a calculator.)
cos (-7π/12)

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Recall the even-odd property of cosine: \(\cos(-\theta) = \cos(\theta)\). So, \(\cos\left(-\frac{7\pi}{12}\right) = \cos\left(\frac{7\pi}{12}\right)\).

Express \(\frac{7\pi}{12}\) as a sum or difference of angles whose cosine and sine values are known. For example, \(\frac{7\pi}{12} = \frac{3\pi}{4} - \frac{\pi}{6}\).

Use the cosine difference identity: \(\cos(a - b) = \cos a \cos b + \sin a \sin b\).

Substitute \(a = \frac{3\pi}{4}\) and \(b = \frac{\pi}{6}\) into the identity: \(\cos\left(\frac{3\pi}{4} - \frac{\pi}{6}\right) = \cos\frac{3\pi}{4} \cos\frac{\pi}{6} + \sin\frac{3\pi}{4} \sin\frac{\pi}{6}\).

Recall the exact values: \(\cos\frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\), \(\cos\frac{\pi}{6} = \frac{\sqrt{3}}{2}\), \(\sin\frac{3\pi}{4} = \frac{\sqrt{2}}{2}\), and \(\sin\frac{\pi}{6} = \frac{1}{2}\). Substitute these to write the expression fully.

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