Question

Use an Addition or

Subtraction Formula to write the expression as a trigonometric

the function of one number, and then find its exact value.

$\cos \frac{13\pi }{15}\cos \left(-\frac{\pi }{5}\right)-\sin \frac{13\pi }{15}\sin \left(-\frac{\pi }{5}\right)$

Short answer

Thus, the exact value of this equation is, $\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}}$.

Step-by-step solution

Step 1. Given information.

The given function is,$\cos \frac{13\pi }{15}\cos \left(-\frac{\pi }{5}\right)-\sin \frac{13\pi }{15}\sin \left(-\frac{\pi }{5}\right)$

Use an addition or subtraction Formula to write the expression.

Then appraise the expression to obtain the final answer.

Step 2. Determining the value.

$\cos \frac{13\pi }{15}\cos \left(-\frac{\pi }{5}\right)-\sin \frac{13\pi }{15}\sin \left(-\frac{\pi }{5}\right)$

This kind of expression appears in the addition of the angles formula for cosine,

$\cos (x+y)=\cos x\cos y-\sin x\sin y$with $x=\frac{13\pi }{15}$and$y=\frac{-\pi }{5}$

Simplify the equation

$\cos \frac{13\pi }{15}\cos \left(-\frac{\pi }{5}\right)-\sin \frac{13\pi }{15}\sin \left(-\frac{\pi }{5}\right)$

$$\begin{gathered} =\cos \left(\frac{13\pi }{15}+\left(\frac{-\pi }{5}\right)\right) \\ =\cos \left(\frac{13\pi }{15}-\frac{3\pi }{15}\right) \\ =\cos \frac{10\pi }{15} \end{gathered}$$