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Chapter 5: Problem 24

Verify each identity. $$\frac{1-\sin \theta}{\cos \theta}=\sec \theta-\tan \theta$$

Short Answer

Expert verified
The given trigonometric identity is verified to be correct. \(\frac{1-\sin \theta}{\cos \theta} = \sec \theta - \tan \theta\).

Step by step solution

01

Write down the problem

The problem to solve is: \(\frac{1-\sin \theta}{\cos \theta} = \sec \theta - \tan \theta\)
02

Start with Left Side

First, start with the left side of the equation. Rewrite it to the form of trigonometric identities. Thus, \(\frac{1-\sin \theta}{\cos \theta} =\frac{1}{\cos \theta}-\frac{\sin \theta}{\cos \theta}\)
03

Apply Trigonometric Identities

Now replace \(\frac{1}{\cos \theta}\) with \(\sec \theta\) and \(\frac{\sin \theta}{\cos \theta}\) with \(\tan \theta\). Now left side becomes \(\sec \theta - \tan \theta\)
04

Compare Both Sides

Since both sides of the identity are the same, that is, \(\sec \theta - \tan \theta = \sec \theta - \tan \theta\), it proves that the given identity is correct.

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Most popular questions from this chapter

Graph: \( f(x)=\frac{5 x^{2}}{x^{2}-25}\) (Section \(2.6, \text { Example } 6)\)

Find the exact value of each expression. Do not use a calculator. $$\cos \left[\cos ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-\sin ^{-1}\left(-\frac{1}{2}\right)\right]$$

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. $$\frac{\sin x}{1-\cos ^{2} x}=\csc x$$

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. $$\sin (x+\pi)=\sin x$$

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