To find \(\cos\left(\frac{5\pi}{12}\right)\), we can use the angle subtraction formula for cosine:

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

Here, we can express \(\frac{5\pi}{12}\) as:

\[
\frac{5\pi}{12} = \frac{4\pi}{12} + \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}
\]

So, \(A = \frac{\pi}{3}\) and \(B = \frac{\pi}{4}\). Plugging these into the formula:

\[
\cos\left(\frac{5\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right)
\]

We know:

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

Substitute these values:

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

Simplify the expression:

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

Thus, the exact value is:

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

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you apply the sine addition formula?
2. Can you derive the exact value for \(\sin\left(\frac{7\pi}{12}\right)\)?
3. How can you express \(\frac{5\pi}{12}\) using different angle identities?
4. What are the values of \(\cos(\frac{3\pi}{12})\) and \(\cos(\frac{7\pi}{12})\)?
5. How does the cosine double angle formula work?

**Tip:** Remember that trigonometric identities can simplify complex expressions into more manageable forms.

Learn how to calculate cos(5pi/12) using the angle subtraction formula for cosine. Detailed step-by-step solution with trigonometric identities explained. Suitable for advanced high school students and anyone interested in trigonometry.

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