Question

use the sum and difference identities to rewrite the following expression as a trigonometric function of one angle.
cosleft(\frac{pi}{6}
ight)cosleft(\frac{3pi}{10}
ight)+sinleft(\frac{pi}{6}
ight)sinleft(\frac{3pi}{10}
ight)

Explanation:

Step1: Recall the cosine difference identity

The cosine - difference identity is $\cos(A - B)=\cos A\cos B+\sin A\sin B$.

Step2: Identify $A$ and $B$

Let $A=\frac{\pi}{6}$ and $B = \frac{3\pi}{10}$.

Step3: Rewrite the expression

Using the cosine - difference identity, $\cos(\frac{\pi}{6})\cos(\frac{3\pi}{10})+\sin(\frac{\pi}{6})\sin(\frac{3\pi}{10})=\cos(\frac{\pi}{6}-\frac{3\pi}{10})$.

Step4: Simplify the angle

First, find a common denominator for the fractions in the angle: $\frac{\pi}{6}-\frac{3\pi}{10}=\frac{5\pi - 9\pi}{30}=\frac{- 4\pi}{30}=-\frac{2\pi}{15}$. Since $\cos(-\theta)=\cos\theta$, the expression is $\cos(\frac{2\pi}{15})$.

Answer:

$\cos(\frac{2\pi}{15})$