Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to determine which of the six trigonometric functions is equal to the expression. Verify your answer algebraically. $$\frac{1}{\sin x}\left(\frac{1}{\cos x}-\cos x\right)$$

Step-by-Step Solution

Answer

The given expression equals to \( \csc x \).

Step 1: Simplify the Expression

First we need to simplify the expression. Start by distributing \( \frac{1}{\sin x} \) inside the parentheses:\[ \frac{1}{\sin x} \left(\frac{1}{\cos x} - \cos x\right) = \frac{1}{\sin x}\cdot \frac{1}{\cos x} - \frac{1}{\sin x} \cdot \cos x = \frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x} \]

Step 2: Use Trigonometric Identities

The next step is to use the quotient of sines and cosines to simplify the expression:\[ \frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x} = \frac{1}{\cos x} \cdot \frac{1}{\sin x} - \frac{\cos x}{\sin x}\] \nIn the first term replace \( \frac{1}{\sin x \cos x} \) with \( \sec x \csc x \), and replace \( \frac{\cos x}{\sin x} \) with \( \cot x \) in the second term:\[ \sec x \csc x - \cot x \]

Step 3: Use Graphing Utility

Now we can use a graphing utility to check which of the six trigonometric functions this expression equals. After plotting the function, we observe that our graph matches with the graph of \( \csc x \). Hence, our expression equals \( \csc x \).

Step 4: Algebraic Verification

Algebraic verification of the solution can be done by remembering that \( \csc x = \frac{1}{\sin x} \). Thus, our expression gives \( \frac{1}{\sin x \cos x} - \frac{\cos x}{\sin x} = \frac{1}{\sin x} - \frac{\cos x}{\sin x} = \csc x - \cot x + \cot x = \csc x \). This confirms that our expression is indeed equal to \( \csc x \).