Chapter 5.1, Problem 39E

To determine

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

Answer to Problem 39E

  $\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }=1$

The result is numerically verified.

Explanation of Solution

Given information:

The given trigonometric expression:

  $\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }$

Formula used:

Reciprocal identity:

  $\sec \alpha =\frac{1}{\cos \alpha }$

Quotient identity:

  $\tan \alpha =\frac{\sin \alpha }{\cos \alpha }$

Calculation:

As per problem,

The given trigonometric expression

  $\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }$

Use the identities: $\sec \alpha =\frac{1}{\cos \alpha }$ and $\tan \alpha =\frac{\sin \alpha }{\cos \alpha }$ to write the expression in terms of sine and cosine.

  $\begin{array}{l}\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }=\frac{1}{\cos \alpha }\cdot \frac{\sin \alpha }{\frac{\sin \alpha }{\cos \alpha }}\text{ [Reciprocal and Quotient identity]}\\ \text{ = }\frac{\sin \alpha \cos \alpha }{\cos \alpha \sin \alpha }\text{ [Simplify]}\\ \text{ =1 [Divide out common factors]}\\ \end{array}$

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 $\text{Y}=\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }$ , which is required to graph.

Step 4: Press GRAPH button to graph the function.

  

From the graph, $\text{Y}=1$ for all points along x-axis.

The table of values shows points (indicated in black in the graph) for the expression.

  

Interpretation:According to the above graph, it can be observed that $\text{Y}=1$ .

Therefore, from the graph, $\sec \alpha \cdot \frac{\sin \alpha }{\tan \alpha }=1$ . Which is same as calculated. Thus, the result is verified.