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Chapter 5: Problem 31

Verify the identity. $$\frac{1}{\cos x+1}+\frac{1}{\cos x-1}=-2 \csc x \cot x$$

Short Answer

Expert verified
Yes, the given identity \(\frac{1}{\cos x+1}+\frac{1}{\cos x-1}=-2 \csc x \cot x\) is valid

Step by step solution

01

Simplify the left-hand side

To simplify the sum on the left-hand side, a common denominator is required, which is \((\cos x+1)(\cos x-1)\). Now, rewrite the sum as \(\frac{\cos x - 1 + \cos x + 1}{(\cos x + 1)(\cos x - 1)}\) which simplifies to \(\frac{2\cos x}{\cos^2 x - 1}\).
02

Use the trigonometric identity \(\cos^2 x = 1 - \sin^2 x\)

Substituting \(\cos^2 x = 1 - \sin^2 x\) in the denominator, \(\frac{2\cos x}{\cos^2 x - 1}\) becomes \(\frac{-2\cos x}{\sin^2 x}\). The negative sign has been taken out of the fraction.
03

Rewrite in terms of other trigonometric relations

The expression can be written in terms of other trigonometric relations. Use the fact that \(\csc x = \frac{1}{\sin x}\) and \(\cot x = \frac{\cos x}{\sin x}\). After substituting these, we get \(-2\csc x \cot x\) as the resulting expression.

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Most popular questions from this chapter

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\csc x+\cot x=1$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x+\tan x-3=0$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\cot ^{2} x-6 \cot x+5=0$$

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x-4 \sec x=0$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the given interval. $$3 \tan ^{2} x+5 \tan x-4=0, \quad\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$
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