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Chapter 5: Problem 31

Verify each identity. $$\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}=2 \sec x$$

Short Answer

Expert verified
After the step by step breakdown, it's been verified that \(\frac{\cos x}{1-\sin x}+\frac{1-\sin x}{\cos x}=2 \sec x\) is indeed a valid trigonometric identity.

Step by step solution

01

Simplify each fraction

Firstly, keep in mind that the main goal is to simplify the equation into the form 2 sec x. Start by breaking the equation into two fractions: cos x/(1-sin x) and (1-sin x)/cos x.
02

Express the fractions as a common fraction

To add fractions, they should have the same denominator. Thus, multiply the first fraction by cos x/cos x and the second fraction by (1-sin x)/(1-sin x), Thus, the equation will be: (cos^2x)/((1-sin x)cos x) + ((1-sin x)^2)/((1-sin x)cos x).
03

Sum the two fractions

Add the two fractions to get a single fraction. This will yield: [cos^2x + (1 - 2sin x + sin^2x)]/((1-sin x)cos x) = (1+ sin^2x - 2sinx)/((1-sin x)cos x).
04

Step 4:Substitute sin^2x = 1 - cos^2x

Using the Pythagorean identity (sin^2 x + cos^2 x = 1 or sin^2 x = 1 - cos^2 x), substitute sin^2 x in the equation to get: (1 + 1 - cos^2x - 2sinx)/((1-sin x)cos x) = (2 - cos^2x - 2sinx)/((1-sin x)cos x).
05

Factor

Factorize the equation to get: 2(1 - cosx + sinx)/((1-sin x)cos x) = 2{(1-cosx)(1+sin x)}/((1-sin x)cos x). The (1-sin x) cancel out to leave: 2/(cosx).
06

Step 6:Convert to sec x

Finally, remember that sec x = 1/cos x. This leads to: 2 sec x. Which matches the right side of the original equation.

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Most popular questions from this chapter

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The most efficient way that I can simplify \(\frac{(\sec x+1)(\sec x-1)}{\sin ^{2} x}\) is to immediately rewrite the expression in terms of cosines and sines.

Find the inverse of \(f(x)=\frac{x-1}{x+1}\) (Section \(1.8, \text { Example } 4)\)

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. I solved \(4 \cos ^{2} x=5-4 \sin x\) by working independently with the left side, applying a Pythagorean identity, and transforming the left side into \(5-4 \sin x\)

Determine whether each -statement makes sense or does not make sense, and explain your reasoning. There are similarities and differences between solving \(4 x+1=3\) and \(4 \sin \theta+1=3:\) In the first equation, I need to isolate \(x\) to get the solution. In the trigonometric equation, I need to first isolate \(\sin \theta,\) but then \(I\) must continue to solve for \(\theta\)

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. $$7 \cos x=4-2 \sin ^{2} x$$
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