Chapter 13.9, Problem 4ST

To determine

To simplify: The trigonometric function.

Answer to Problem 4ST

The $2\sin 5\alpha \cos 5\alpha =\sin \left(10\alpha \right)$ .

Explanation of Solution

Given information:

The given information is $2\sin 5\alpha \cos 5\alpha$ .

Calculation:

Consider,

  $2\sin 5\alpha \cos 5\alpha$ .

Now simplify the above trigonometric function of a single angle.

In general the addition formula for the sine

  $sin\left(\alpha +\beta \right)=sin\alpha cos\beta +cos\alpha sin\beta$

Thus,

  $\begin{array}{l}2\sin 5\alpha \cos 5\alpha =\sin 5\alpha \cos 5\alpha +\sin 5\alpha \cos 5\alpha \\ 2\sin 5\alpha \cos 5\alpha =\sin \left(5\alpha +5\alpha \right)\\ 2\sin 5\alpha \cos 5\alpha =\sin \left(10\alpha \right)\end{array}$

Therefore, the $2\sin 5\alpha \cos 5\alpha =\sin \left(10\alpha \right)$ .