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Chapter 5: Problem 5

Verify each identity. $$\tan x \csc x \cos x=1$$

Short Answer

Expert verified
The given expression simplifies to 1, which verifies the identity.

Step by step solution

01

Substitute Trigonometric Identities

We substitute \(\tan x\) with \(\sin x / \cos x\) and \(\csc x\) with \(1 / \sin x\), the given identity becomes \((\sin x / \cos x) * (1 / \sin x) * \cos x\).
02

Simplify the Expression

We simplify the equation by cancelling out similar terms. Here, \(\sin x\) in the numerator cancels out with \(\sin x\) in the denominator, and \(\cos x\) in the denominator cancels out with \(\cos x\) in the numerator. We get 1, which is the right side.

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Most popular questions from this chapter

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. $$\cos ^{2} x+5 \cos x-1=0$$

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. $$15 \cos ^{2} x+7 \cos x-2=0$$

Solve each equation on the interval \([0,2 \pi)\) Do not use a calculator. $$2 \cos x-1+3 \sec x=0$$

Determine the amplitude, period, and phase shift of \(y=4 \sin (2 \pi x+2) .\) Then graph one period of the function. (Section 4.5, Example 4 )

Verify the identity: $$\frac{\sin (x-y)}{\cos x \cos y}+\frac{\sin (y-z)}{\cos y \cos z}+\frac{\sin (z-x)}{\cos z \cos x}=0$$
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