find the exact value of cos(7π/12). a. (√2 - √6)/4 b. (√6 - √2)/4 c. (√2 - √6)/2 d. (2 - √6)/4 please select…

Explanation:

Step1: Rewrite the angle

We know that $\frac{7\pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$. Then, by the cosine - sum formula $\cos(A + B)=\cos A\cos B-\sin A\sin B$, where $A=\frac{\pi}{3}$ and $B = \frac{\pi}{4}$.

Step2: Find the values of trigonometric functions

We know that $\cos\frac{\pi}{3}=\frac{1}{2}$, $\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$, $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, and $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$.

Step3: Apply the formula

$\cos(\frac{\pi}{3}+\frac{\pi}{4})=\cos\frac{\pi}{3}\cos\frac{\pi}{4}-\sin\frac{\pi}{3}\sin\frac{\pi}{4}=\frac{1}{2}\times\frac{\sqrt{2}}{2}-\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}=\frac{\sqrt{2}-\sqrt{6}}{4}$.

Answer:

A. $\frac{\sqrt{2}-\sqrt{6}}{4}$