Chapter 3.8, Problem 1P

To determine

To rewrite: The expression $\cos 9t-\cos 7t$ as a product of two trigonometric functions of different frequencies.

Answer to Problem 1P

The expression $\cos 9t-\cos 7t$ can be rewritten as $\underset{\_}{\cos 9t-\cos 7t=-2\sin 8t\sin t}$.

Explanation of Solution

Formula used:

The known trigonometric identity is that $\cos A-\cos B=-2\sin \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)$.

Calculation:

The given expression is $\cos 9t-\cos 7t$.

Apply the above formula by substituting $A=9t$ and $B=7t$ as follows.

$\begin{array}{c}\cos 9t-\cos 7t=-2\sin \left(\frac{9t+7t}{2}\right)\sin \left(\frac{9t-7t}{2}\right)\\ =-2\sin \left(\frac{16t}{2}\right)\sin \left(\frac{2t}{2}\right)\\ =-2\sin 8t\sin t\end{array}$

Thus, the expression $\cos 9t-\cos 7t$ can be rewritten as $\underset{\_}{\cos 9t-\cos 7t=-2\sin 8t\sin t}$.