Chapter 5.5, Problem 30E

To determine

To write:The expression $10{\sin }^{2}x-5$ with the help of double angle formula and verify both the expressions by graphing utility.

Answer to Problem 30E

The expression $10{\sin }^{2}x-5$ can be written as $-5\cos 2x$ with the help of double angle formula.

Explanation of Solution

Given information:

The expression is given as $10{\sin }^{2}x-5$ .

Formula used:

Use the double angle formula ${\sin }^{2}x=\frac{1-\cos 2x}{2}$ .

Calculation:

As per the double angle formula.

  ${\sin }^{2}x=\frac{1-\cos 2x}{2}$

Substitute the value in the above expression.

  $\begin{array}{c}10{\sin }^{2}x-5=10\left(\frac{1-\cos 2x}{2}\right)-5\\ =5-5\cos 2x-5\\ =-5\cos 2x\end{array}$

Verify both the expressions using graphical utility:

Draw the graph for the expression $10{\sin }^{2}x-5$ .

  

Figure (1)

Draw the graph for the expression $-5\cos 2x$ .

  

Figure (2)

As both the expressions shows same values in Figure (1) and Figure (2) at each point.

Therefore both the expressions $10{\sin }^{2}x-5$ and $-5\cos 2x$ are same.