Chapter 5.5, Problem 29E

To determine

To write:The expression ${\cos }^{2}x-\frac{1}{2}$ with the help of double angle formula and verify both the expressions by graphing utility.

Answer to Problem 29E

The expression ${\cos }^{2}x-\frac{1}{2}$ can be written as $\frac{\cos 2x}{2}$ with the help of double angle formula.

Explanation of Solution

Given information:

The expression is given as ${\cos }^{2}x-\frac{1}{2}$ .

Formula used:

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

Calculation:

As per the double angle formula.

  ${\cos }^{2}x=\frac{1+\cos 2x}{2}$

Substitute the value in the above expression.

  $\begin{array}{c}{\cos }^{2}x-\frac{1}{2}=\frac{1+\cos 2x}{2}-\frac{1}{2}\\ =\frac{1+\cos 2x-1}{2}\\ =\frac{\cos 2x}{2}\end{array}$

Verify both the expressions using graphical utility:

Draw the graph for the expression ${\cos }^{2}x-\frac{1}{2}$ .

  

Figure (1)

Draw the graph for the expression $\frac{\cos 2x}{2}$ .

  

Figure (2)

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

Therefore both the expressions ${\cos }^{2}x-\frac{1}{2}$ and $\frac{\cos 2x}{2}$ are same.