Find the exact value of the expression.
$\(\cos\]\frac{5\pi}{12}\)$

• A. $\(\frac{\sqrt2-\sqrt6}{4}\)$
• B. $\(\frac{\sqrt6-\sqrt2}{4}\)$
• C. $\(\frac{\sqrt2-\sqrt6}{2}\)$
• D. $\(\frac{\sqrt4}{4}\)$

Answer: (B)

Verified step by step guidance

1. First, recognize that \( \cos\frac{5\pi}{12} \) is an angle that can be expressed in terms of known angles. We can use the angle sum or difference identities to find its exact value.
2. Express \( \frac{5\pi}{12} \) in terms of angles whose cosine values are known. Notice that \( \frac{5\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \). This allows us to use the cosine difference identity: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).
3. Substitute \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \) into the identity: \( \cos\left(\frac{\pi}{3} - \frac{\pi}{4}\right) = \cos\frac{\pi}{3} \cos\frac{\pi}{4} + \sin\frac{\pi}{3} \sin\frac{\pi}{4} \).
4. Calculate the known values: \( \cos\frac{\pi}{3} = \frac{1}{2} \), \( \cos\frac{\pi}{4} = \frac{\sqrt{2}}{2} \), \( \sin\frac{\pi}{3} = \frac{\sqrt{3}}{2} \), and \( \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2} \).
5. Substitute these values into the expression: \( \cos\frac{5\pi}{12} = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right) \). Simplify the expression to find the exact value.