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Chapter 5: Problem 52

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

Short Answer

Expert verified
The trigonometric function that the given expression simplifies to is \( \tan(x) \sec(x) \) which was algebraically verified.

Step by step solution

01

Simplify the Expression

The expression given is \(\frac{1}{\sin x}(\frac{1}{\cos x}-\cos x)\). This can be simplified by reducing the fractions inside the bracket. The result is \(\frac{1}{\sin x} (\sec x-\cos x)\), where \( \sec x = \frac{1}{\cos x} \).
02

Factor Out Common Term

To further simplify the expression, factor out the common term, which in this case would be \(\sec x\). The expression then becomes \(\frac{1}{\sin x} \sec x(1 - \cos^2 x)\). We know the identity \(1 - cos^2(x) = \sin^2(x)\), hence we can substitute, resulting in \(\frac{1}{\sin x} \sec x (\sin^2 x)\).
03

Simplify Further and Identify The Function

Simplifying the expression leads to \(\frac{\sec x \sin x}{\cos x}\). Now, note that \(\sec x = \frac{1}{\cos x}\), so \(\frac{\sec x}{\cos x} = \frac{1}{\cos^2 x}\). Therefore, the expression simplifies to \(\frac{\sin x}{\cos^2 x}\). This is equivalent to \( \sin x \sec^2 x \) which also fulfills the definition of \( \tan(x) \sec(x) \).
04

Verification

One can verify this by choosing special values like \( x=\pi/4 \) and checking whether both the original and the final expression admit the same values. For \( \tan(x) \sec(x) \), inserting \( x=\pi/4 \) the result is 1, the same as for the initial expression proving the correctness of the simplification.

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