Chapter 7.4, Problem 34E

To determine

To verify: that the given equation is an identity.

Explanation of Solution

Given information:

  $\sin 3x=3\sin x-4{\sin }^{3}x$

Calculation:

Simplify the left side of the equation into right side as follows:

  $\begin{array}{c}\sin 3x=\sin \left(2x+x\right)\end{array}$

  $\begin{array}{c}=\sin 2x\cos x+\sin x\cos 2x\left[\text{Sum identity of sine}\right]\end{array}$

  $\begin{array}{c}=\left(2\sin x\cos x\right)\cos x+\sin x\left(1-2{\sin }^{2}x\right)\left[\text{Double angle identity of sine and cosine}\right]\end{array}$

  $\begin{array}{c}=2\sin x{\cos }^{2}x+\sin x-2{\sin }^{3}x\end{array}$

  $\begin{array}{c}=2\sin x\left(1-{\sin }^{2}x\right)+\sin x-2{\sin }^{3}x\left[\text{Pythagorean identity}\right]\end{array}$

  $\begin{array}{c}=2\sin x-2{\sin }^{3}x+\sin x-2{\sin }^{3}x\end{array}$

  $\begin{array}{c}=3\sin x-4{\sin }^{3}x\end{array}$

Since the left side of the equation is transformed into right hand side. So, the equation is an identity.