To find \(\cos\left(\frac{5\pi}{12}\right)\) using the exact values of \(\frac{\pi}{4}\) and \(\frac{\pi}{6}\), let's rewrite \(\frac{5\pi}{12}\) as a sum of these angles:

\[
\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}
\]

Now, use the cosine addition formula:

\[
\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)
\]

Here, \(A = \frac{\pi}{4}\) and \(B = \frac{\pi}{6}\). Substituting these into the formula:

\[
\cos\left(\frac{5\pi}{12}\right) = \cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{4}\right)\sin\left(\frac{\pi}{6}\right)
\]

Using the known exact values:

\[
\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}, \quad \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}
\]

Substitute these values into the equation:

\[
\cos\left(\frac{5\pi}{12}\right) = \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right)
\]

Simplify the expression:

\[
\cos\left(\frac{5\pi}{12}\right) = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

So, the exact value is:

\[
\boxed{\frac{\sqrt{6} - \sqrt{2}}{4}}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How can you use the sine subtraction formula to find \(\sin\left(\frac{5\pi}{12}\right)\)?
2. Can you derive the exact value for \(\cos\left(\frac{7\pi}{12}\right)\)?
3. What is the difference between the sine and cosine addition formulas?
4. How can you express \(\frac{7\pi}{12}\) using different angle identities?
5. What are the exact values for \(\sin(\frac{\pi}{3})\) and \(\cos(\frac{\pi}{3})\)?

**Tip:** When using trigonometric identities, always verify which identities are most appropriate for simplifying the given expression.

Learn how to find the exact value of cos(5π/12) using the cosine addition formula and exact trigonometric values for π/4 and π/6. This problem involves advanced trigonometric identities and is suitable for advanced high school students.

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