Chapter 5.1, Problem 44E

To determine

To simplify: The given trigonometric expression and check the result using the table feature of a graphing utility.

Answer to Problem 44E

  $\frac{1}{{\cot }^{2}x+1}={\sin }^{2}x$

The result is numerically verified from the graph.

Explanation of Solution

Given information:

The given trigonometric expression:

  $\frac{1}{{\cot }^{2}x+1}$

Formula used:

The following Pythagorean identity:

  ${\cot }^{2}x+1={\csc }^{2}x$

Calculation:

As per problem,

The given trigonometric expression

  $\frac{1}{{\cot }^{2}x+1}$

Use the Pythagorean identity, ${\cot }^{2}x+1={\csc }^{2}x$ , to simplify. $\begin{array}{l}\frac{1}{{\cot }^{2}x+1}=\frac{1}{{\csc }^{2}x}\text{ [Pythagorean identity]}\\ \text{ = }{\left(\frac{1}{\csc x}\right)}^2\text{ [Rule of exponents]}\\ \text{ = }{\left(\sin x\right)}^2\text{ [Reciprocal identity:}\sin x=\frac{1}{\csc x}\text{]}\\ {\text{ = sin}}^{2}x\text{ [Rule of exponents]}\\ \end{array}$

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 ${\text{Y}}_1=\frac{1}{{\cot }^{2}x+1}$ and ${\text{Y}}_2={\sin }^{2}x$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

The curve of ${\text{Y}}_1$ is indicated by yellow line and the curve of ${\text{Y}}_2$ is indicated by green line. Both the curves coincide.

The table shows some of the points (indicated in green in the graph) that coincide for both the expressions.

  

Interpretation: According to the above graph, it can be observed that ${\text{Y}}_1={\text{Y}}_2$ as both the graphs coincide.

Therefore, from the graph, $\frac{1}{{\cot }^{2}x+1}={\sin }^{2}x$ , which is same as calculated. Thus, the result is verified.