use an addition or subtraction formula to write the expression as a trigonometric function of one number…

Explanation:

Step1: Recall cosine - difference formula

The formula $\cos(A - B)=\cos A\cos B+\sin A\sin B$. Here, $A = \frac{4\pi}{5}$ and $B=\frac{7\pi}{15}$.

Step2: Calculate $A - B$

$A - B=\frac{4\pi}{5}-\frac{7\pi}{15}=\frac{12\pi - 7\pi}{15}=\frac{5\pi}{15}=\frac{\pi}{3}$. So, $\cos(\frac{4\pi}{5})\cos(\frac{7\pi}{15})+\sin(\frac{4\pi}{5})\sin(\frac{7\pi}{15})=\cos(\frac{4\pi}{5}-\frac{7\pi}{15})=\cos(\frac{\pi}{3})$.

Step3: Find the exact value

We know that $\cos(\frac{\pi}{3})=\frac{1}{2}$.

Answer:

$\frac{1}{2}$