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Chapter 5: Problem 20

Verify the identity. $$\frac{1}{\tan \beta}+\tan \beta=\frac{\sec ^{2} \beta}{\tan \beta}$$

Short Answer

Expert verified
The identity \(\frac{1}{\tan \beta}+\tan \beta=\frac{\sec ^{2} \beta}{\tan \beta}\) has been verified successfully after converting the terms to sines and cosines, cross multiplying and simplifying the equation

Step by step solution

01

- Convert Everything to Sines and Cosines

The conversion is done by replacing \(\tan \beta = \frac{\sin \beta}{\cos \beta}\) and \(\sec \beta = \frac{1}{\cos \beta}\), which gives \(\frac{\cos \beta}{\sin \beta}+\frac{\sin \beta}{\cos \beta}=\frac{1}{\cos^{2} \beta \sin \beta}\)
02

- Cross multiply the terms

Cross multiply the terms \((\cos^2 \beta + \sin^2 \beta) = \frac{1}{\cos^2 \beta \sin \beta}\) which simplifies to \(\cos^2 \beta + \sin^2 \beta = \frac{1}{\cos^2 \beta \sin \beta}\)
03

- Utilize Pythagorean Identity

The Pythagorean Identity states that \(\sin^2 \beta + \cos^2 \beta = 1\). In place of \(\sin^2 \beta + \cos^2 \beta\) we write 1. After this substitution, the equation simplifies to \(1 = \frac{1}{\cos^2 \beta \sin \beta}\)
04

- Simplify the Equation

After simplification, the equation will become \(1 = \frac{1}{\cos^2 \beta \sin \beta}\) which means the left-hand side equals the right-hand side, confirming the identity

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Most popular questions from this chapter

Find the exact values of the sine, cosine, and tangent of the angle. $$\frac{7 \pi}{12}=\frac{\pi}{3}+\frac{\pi}{4}$$

Use a graphing utility to approximate the solutions (to three decimal places) of the equation in the interval \([0,2 \pi)\). $$x \tan x-1=0$$

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