use the sum and difference identities to rewrite the following expression as a trigonometric function of one…

use the sum and difference identities to rewrite the following expression as a trigonometric function of one angle.\n$$\\cos\\left(\\frac{3\\pi}{10}\\right)\\cos\\left(\\frac{4\\pi}{5}\\right) + \\sin\\left(\\frac{3\\pi}{10}\\right)\\sin\\left(\\frac{4\\pi}{5}\\right)$$

use the sum and difference identities to rewrite the following expression as a trigonometric function of one angle.\n$$\\cos\\left(\\frac{3\\pi}{10}\\right)\\cos\\left(\\frac{4\\pi}{5}\\right) + \\sin\\left(\\frac{3\\pi}{10}\\right)\\sin\\left(\\frac{4\\pi}{5}\\right)$$

Answer

Explanation:

Step1: Identify the relevant trigonometric identity

The expression matches the cosine difference identity: $\cos(A)\cos(B) + \sin(A)\sin(B) = \cos(A - B)$.

Step2: Assign values to variables A and B

Let $A = \frac{3\pi}{10}$ and $B = \frac{4\pi}{5}$.

Step3: Substitute values into the identity

$$\cos\left(\frac{3\pi}{10} - \frac{4\pi}{5}\right)$$

Step4: Find a common denominator for the angles

$$\cos\left(\frac{3\pi}{10} - \frac{8\pi}{10}\right)$$

Step5: Subtract the fractions

$$\cos\left(-\frac{5\pi}{10}\right)$$

Step6: Simplify the fraction and apply parity

Since $\cos(-x) = \cos(x)$, we have $\cos\left(-\frac{\pi}{2}\right) = \cos\left(\frac{\pi}{2}\right)$.

Answer:

$\cos\left(\frac{\pi}{2}\right)$