Chapter 7.3, Problem 98E

(a)

To determine

To graph: the function $f\left(x\right)=\cos 2x+2{\sin }^{2}x$ and make a conjecture.

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\cos 2x+2{\sin }^{2}x$

Graph:

Now, consider the function.

  $f\left(x\right)=\cos 2x+2{\sin }^{2}x$

Sketch the graph of the function using graphing utility.

  

Interpretation:

From the graph it can be concluded that the function is a horizontal line parallel to x-axis and it equation is $y=1$ .

Hence, the conjecture can be made that the function is given as

  $f\left(x\right)=1$ .

(b)

To determine

To verify: the conjecture of part (a).

Explanation of Solution

Given information:

Given function

  $f\left(x\right)=\cos 2x+2{\sin }^{2}x$

Calculation:

Now, consider the function.

  $f\left(x\right)=\cos 2x+2{\sin }^{2}x$

The conjecture from part (a) is $f\left(x\right)=1$ .

Use the identity

  $\cos 2x=1-2{\sin }^{2}x$ to simplify the expression $f\left(x\right)=\cos 2x+2{\sin }^{2}x$ .

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

Hence, the conjecture is verified.