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Let's solve each part of the problem step-by-step by finding the exact values of the expressions involving cosine sums and differences.

### Part (a): \(\cos\left(\frac{5\pi}{12}\right) + \cos\left(\frac{\pi}{12}\right)\)

We use the sum-to-product formula for cosines:
\[
\cos A + \cos B = 2\cos\left(\frac{A+B}{2}\right)\cos\left(\frac{A-B}{2}\right)
\]
where \( A = \frac{5\pi}{12} \) and \( B = \frac{\pi}{12} \).

1. Calculate \( A + B \):
   \[
   A + B = \frac{5\pi}{12} + \frac{\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2}
   \]

2. Calculate \( A - B \):
   \[
   A - B = \frac{5\pi}{12} - \frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3}
   \]

3. Plug into the sum-to-product formula:
   \[
   \cos\left(\frac{5\pi}{12}\right) + \cos\left(\frac{\pi}{12}\right) = 2\cos\left(\frac{\pi}{4}\right)\cos\left(\frac{\pi}{6}\right)
   \]

4. Use known values:
   \[
   \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
   \]

Thus, the final expression becomes:
\[
2 \times \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}
\]

### Part (b): \(\cos\left(\frac{5\pi}{24}\right) - \cos\left(\frac{\pi}{24}\right)\)

We use the difference-to-product formula:
\[
\cos A - \cos B = -2\sin\left(\frac{A+B}{2}\right)\sin\left(\frac{A-B}{2}\right)
\]
where \( A = \frac{5\pi}{24} \) and \( B = \frac{\pi}{24} \).

1. Calculate \( A + B \):
   \[
   A + B = \frac{5\pi}{24} + \frac{\pi}{24} = \frac{6\pi}{24} = \frac{\pi}{4}
   \]

2. Calculate \( A - B \):
   \[
   A - B = \frac{5\pi}{24} - \frac{\pi}{24} = \frac{4\pi}{24} = \frac{\pi}{6}
   \]

3. Plug into the difference-to-product formula:
   \[
   \cos\left(\frac{5\pi}{24}\right) - \cos\left(\frac{\pi}{24}\right) = -2\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{\pi}{12}\right)
   \]

4. Use known values for \( \sin\left(\frac{\pi}{4}\right) \) and \( \sin\left(\frac{\pi}{4}\right) \):
   
Thus, the final expression becomes:
\[
= -2
= 2


Find the exact value of the following expressions: (a) cos(5π/12) + cos(π/12), (b) cos(5π/24) - cos(π/24).

Learn how to find the exact values of trigonometric expressions such as cos(5π/12) + cos(π/12) and cos(5π/24) - cos(π/24) using the sum-to-product and difference-to-product formulas. This detailed solution explains each step and simplifies the result using known trigonometric values.

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