Question

use identities to find the exact value.

$\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$

a. $- 1$
b. $\frac { \sqrt { 3 } } { 2 }$
c. $\frac { 1 } { 2 }$
d. $\frac { \sqrt { 2 } } { 2 }$

Explanation:

Step1: Use the cosine difference identity

The identity \(\cos(A - B)=\cos A\cos B+\sin A\sin B\). Let \(A=\frac{7\pi}{12}\) and \(B = \frac{5\pi}{12}\). Then \(\cos\frac{7\pi}{12}\cos\frac{5\pi}{12}+\sin\frac{7\pi}{12}\sin\frac{5\pi}{12}=\cos(\frac{7\pi}{12}-\frac{5\pi}{12})\).

Step2: Simplify the expression inside the cosine function

\(\frac{7\pi}{12}-\frac{5\pi}{12}=\frac{7\pi - 5\pi}{12}=\frac{2\pi}{12}=\frac{\pi}{6}\).

Step3: Evaluate the cosine function

We know that \(\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}\).

Answer:

B. \(\frac{\sqrt{3}}{2}\)