Chapter 5.5, Problem 105E

To determine

To verify: The given identity.

Explanation of Solution

Given information:

The following identity:

  $\frac{\sin 3y}{\sin y}=1-2{\sin }^{2}y+2{\cos }^{2}y$

Formula used:

The following sum formula:

  $\sin (u+v)=\sin u\cos v+\cos u\sin v$

The following double angle formulas:

  $\begin{array}{l}\sin 2u=2\sin u\cos u\\ \cos 2u=1-2{\sin }^{2}u\end{array}$

Proof:

The given identity is verified as follows:

Working with the left side,

  $\begin{array}{l}\frac{\sin 3y}{\sin y}=\frac{\sin (2y+y)}{\sin y}\text{ [Expand]}\\ =\frac{\sin 2y\cos y+\cos 2y\sin y}{\sin y}\text{ [Sum formula]}\\ =\frac{(2\sin y\cos y)\cos y+(1-2{\sin }^{2}y)\sin y}{\sin y}\text{ [Double angle formula]}\\ =\frac{2\sin y{\cos }^{2}y+\sin y-2{\sin }^{3}y}{\sin y}\text{ [Multiply]}\\ =\frac{\sin y(2{\cos }^{2}y+1-2{\sin }^{2}y)}{\sin y}\text{ [Factor]}\\ =1-2{\sin }^{2}y+2{\cos }^{2}y\text{ [Simplify]}\end{array}$

Since both sides are equal, the identity is verified

The identity is verified graphically as:

Graph:

Sketch the graph using graphing utility.

Step 1: Press WINDOW button to access the Window editor.

Step 2: Press $\boxed{\text{Y=}}$ button.

Step 3: Enter the expressions ${\text{Y}}_1=\frac{\sin 3y}{\sin y}$ and ${\text{Y}}_2=1-2{\sin }^{2}y+2{\cos }^{2}y$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

Since both the curves coincide, the identity is verified.