Chapter 5.1, Problem 79E

To determine

To graph: The given trigonometric expression and determine which trigonometric function is equal to the expression. Then verify the resulting identity algebraically.

Explanation of Solution

Given information:

The following trigonometric expression:

  $\frac{1}{\sin x}(\frac{1}{\cos x}-\cos x)$

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 expression $\frac{1}{\sin x}(\frac{1}{\cos x}-\cos x)$ which is required to graph.

Step 4: Press GRAPH button to graph the function.

The graph is obtained as:

  

Interpretation:

According to the above graph, it can be observed that the given expression has the same graph as obtained for tangent function. Hence, $y=\tan x$

Formula used:

The following identities:

The Pythagorean identity:

  ${\sin }^{2}x=1-{\cos }^{2}x$

The Quotient identity:

  $\tan x=\frac{\sin x}{\cos x}$

Proof:

The given trigonometric expression can be simplified as follows:

  $\begin{array}{l}y=\frac{1}{\sin x}(\frac{1}{\cos x}-\cos x)\text{ [Original expression]}\\ =\frac{1}{\sin x}(\frac{1-{\cos }^{2}x}{\cos x})\text{ [Subtract fractions]}\\ =\frac{1}{\sin x}(\frac{{\sin }^{2}x}{\cos x})\text{ [Pythagorean identity]}\\ =\frac{\sin x}{\cos x}\text{ [Divide out common factors]}\\ =\tan x\text{ [Quotient identity]}\end{array}$

Therefore, $\text{y}=\tan x$ .

The simplified expression is same as the expression obtained from the graph. Hence, verified.