Chapter 7, Problem 30RE

(a)

To determine

To draw:The graph of the functions $f\left(x\right)=\sin x\cot \frac{x}{2}$ and $g\left(x\right)=\cos x$ , and make a conjecture.

Answer to Problem 30RE

Thegraph of the functions $f\left(x\right)=\sin x\cot \frac{x}{2}$ and $g\left(x\right)=\cos x$ is shown in the figure (1) and conjecture is $\sin x\cot \frac{x}{2}=\cos x+1$ .

Explanation of Solution

Given information:

The functions are $f\left(x\right)=\sin x\cot \frac{x}{2}$ and $g\left(x\right)=\cos x$ .

Graph:

Plot the graph of the functions $f\left(x\right)=\sin x\cot \frac{x}{2}$ and $g\left(x\right)=\cos x$ as follows.

  

Figure (1)

Interpretation:

In the above graph, a reasonable conjecture will be $f\left(x\right)=g\left(x\right)+1$ because every point of the solid graph is one unit above the corresponding point on the dashed graph.

Therefore, the graph of the functions $f\left(x\right)=\sin x\cot \frac{x}{2}$ and $g\left(x\right)=\cos x$ is shown in the figure (1) and conjecture is $\sin x\cot \frac{x}{2}=\cos x+1$ .

(b)

To determine

To prove:The conjecture is $2{\sin }^{2}3x+\cos 6x=1$ .

Explanation of Solution

Given information:

The conjecture is $\sin x\cot \frac{x}{2}=\cos x+1$ .

Calculation:

Simplify the function.

  $\begin{array}{c}\sin x\cot \frac{x}{2}=\cos x+1\\ \sin x\left(\frac{1+\cos x}{\sin x}\right)=\cos x+1\\ \cos x+1=\cos x+1\end{array}$

From the above, it is clear that the left hand side is equal to the right hand side.

Hence, proved.