Let's tackle the problem step by step:
### Part a. Identify [tex]$\alpha$[/tex] and [tex]$\beta$[/tex] in the expression.
Given the expression:
[tex]\[
\cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right)
\][/tex]
we know this matches the cosine of the difference formula:
[tex]\[
\cos(\alpha - \beta) = \cos(\alpha)\cos(\beta) + \sin(\alpha)\sin(\beta)
\][/tex]
By comparing, we identify:
[tex]\[
\alpha = \frac{7\pi}{12}
\][/tex]
[tex]\[
\beta = \frac{5\pi}{12}
\][/tex]
So:
[tex]\[
\text{The value of } \alpha \text{ is } \frac{7\pi}{12}
\][/tex]
[tex]\[
\text{The value of } \beta \text{ is } \frac{5\pi}{12}
\][/tex]
### Part b. Write the expression as the cosine of an angle.
The expression can be rewritten using the cosine of the difference formula:
[tex]\[
\cos \left(\frac{7\pi}{12} - \frac{5\pi}{12}\right)
\][/tex]
Simplifying the angle inside the cosine function:
[tex]\[
\frac{7\pi}{12} - \frac{5\pi}{12} = \frac{2\pi}{12} = \frac{\pi}{6}
\][/tex]
Therefore, the expression simplifies to:
[tex]\[
\cos \left(\frac{\pi}{6}\right)
\][/tex]
So:
[tex]\[
\text{The expression is } \cos \left(\frac{\pi}{6}\right)
\][/tex]
### Part c. Find the exact value of the expression.
We know from trigonometric identities that:
[tex]\[
\cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}
\][/tex]
Thus, the exact value of the initial expression is:
[tex]\[
\cos \left(\frac{7\pi}{12}\right) \cos \left(\frac{5\pi}{12}\right) + \sin \left(\frac{7\pi}{12}\right) \sin \left(\frac{5\pi}{12}\right) = \frac{\sqrt{3}}{2}
\][/tex]
So:
[tex]\[
\cos \left(\frac{7 \pi}{12}\right) \cos \left(\frac{5 \pi}{12}\right) + \sin \left(\frac{7 \pi}{12}\right) \sin \left(\frac{5 \pi}{12}\right) = \frac{\sqrt{3}}{2}
\][/tex]
