Chapter 5.1, Problem 48E

To determine

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

Answer to Problem 48E

  $\frac{{\csc }^{2}x-1}{\csc x-1}=\csc x+1$

The result is graphically verified as the curve of the given expression coincides at all points with the graph of its result.

Explanation of Solution

Given information:

The given trigonometric expression:

  $\frac{{\csc }^{2}x-1}{\csc x-1}$

Formula used:

The difference of two squares can be factorized as:

  $(a^2-b^2)=(a+b)(a-b)$

Calculation:

As per problem,

The given trigonometric expression

  $\frac{{\csc }^{2}x-1}{\csc x-1}$

Write the numerator in factor form using $(a^2-b^2)=(a+b)(a-b)$ formula.

  $\begin{array}{l}\frac{{\csc }^{2}x-1}{\csc x-1}=\frac{{(\csc x)}^2-{(1)}^2}{\csc x-1}\text{ [Rule of exponents]}\\ \text{ = }\frac{(\csc x+1)(\csc x-1)}{\csc x-1}\text{ [Use (}{a}^2-b^2\text{) formula]}\\ \text{ = }\csc x+1\text{ [Divide out common factor]}\\ \end{array}$

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 ${\text{Y}}_1=\frac{{\csc }^{2}x-1}{\csc x-1}$ and ${\text{Y}}_2=\csc x+1$ , 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 and table, it can be observed that ${\text{Y}}_1={\text{Y}}_2$ as both the graphs coincide.

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