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Chapter 5: Problem 71

Rewrite each expression in terms of the given function or functions. $$\frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} ; \csc x$$

Short Answer

Expert verified
\(\frac{1}{\csc x} - \sqrt{1-\frac{1}{\csc ^2 x}}\)

Step by step solution

01

Understand the Trigonometric Identity

The key trigonometric identity to use in this situation is \( \csc^2 x = 1 + \cot^2 x \). This identity allows to rewrite any expression involving cosecant squared in terms of cotangent squared or vice versa.
02

Express Given Function in terms of Sine and Cosine

To start off, express each part of the expression in terms of sine and cosine. As \( \csc x = \frac{1}{\sin x} \) - this will be helpful later on. So the expression becomes \( \frac{1}{1-\cos x}-\frac{\cos x}{1+\cos x} = \frac{\sin^2 x}{\sin x(1-\cos x)}-\frac{\sin x \cos x}{\sin x(1+\cos x)} \)
03

Simplifying the expression

The resulting expression can now be combined over a common denominator and simplified. It reduces to \( \frac{\sin^2 x - \sin x \cos x}{\sin x} = \frac{\sin x (\sin x - \cos x)}{\sin x} = \sin x - \cos x \). This can be further simplified using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \) to rewrite the \( \cos x \) term as \( \sin x - \sqrt{(1-\sin^2 x)} \)
04

Rewrite in terms of cosecant

Finally, substitute \( \sin x \) with \( \frac{1}{\csc x} \). After substitution, the expression becomes \( \frac{1}{\csc x} - \sqrt{1-\frac{1}{\csc ^2 x}} \). This is the final expression in terms of \( \csc x \).

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Most popular questions from this chapter

Exercises \(116-118\) will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$\cos \frac{\pi}{2} \cos \frac{\pi}{3}=\frac{1}{2}\left[\cos \left(\frac{\pi}{2}-\frac{\pi}{3}\right)+\cos \left(\frac{\pi}{2}+\frac{\pi}{3}\right)\right]$$

Use a graphing utility to approximate the solutions of each equation in the interval \([0,2 \pi) .\) Round to the nearest hundredth of a radian. $$15 \cos ^{2} x+7 \cos x-2=0$$

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\sin (x+\pi)=\sin x$$

Describe a general strategy for solving each equation. Do not solve the equation. $$\sin 2 x=\sin x$$

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$3 \tan ^{2} x-\tan x-2=0$$
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