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Chapter 5: Problem 38

Verify each identity. $$\frac{\tan ^{2} x-\cot ^{2} x}{\tan x+\cot x}=\tan x-\cot x$$

Short Answer

Expert verified
The identity \(\frac{\tan^{2} x - \cot^{2} x}{\tan x + \cot x} = \tan x - \cot x\) is verified.

Step by step solution

01

Write the Definitions

Rewrite the given expression with the definitions of \( \tan x \) and \( \cot x \). Remember, \( \tan x = \frac{\sin x}{\cos x} \) and \( \cot x = \frac{\cos x}{\sin x} \). The expression becomes \(\frac{(\frac{\sin^{2} x}{\cos^{2} x} - \frac{\cos^{2} x}{\sin^{2} x})}{\frac{\sin x}{\cos x} + \frac{\cos x}{\sin x}} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\).
02

Simplify the Expression

Simplify the numerator and the denominator of the fraction in the left-hand side (LHS). The simplification gives: \(\frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x \cos^{2} x}\div \frac{\sin^{2} x + \cos^{2} x}{\sin x \cos x}\).
03

Further Simplify the Expression

Further simplification gives: \(\frac{\sin^{4} x - \cos^{4} x}{\sin^{2} x + \cos^{2} x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\), which will be recognized as the difference of squares \(\sin^{4} x - \cos^{4} x =(\sin^{2} x + \cos^{2} x)(\sin^{2} x - \cos^{2} x)\). Then cancel out the common factors.
04

Final Simplification

After cancellation, it is observed that, \(\frac{\sin^{2} x - \cos^{2} x}{\sin x \cos x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\). This implies \(\frac{\sin x}{\cos x} - \frac{\cos x}{\sin x} = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}\), and thus the identity is verified.

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Most popular questions from this chapter

Remembering the six sum and difference identities can be difficult. Did you have problems with some exercises because the identity you were using in your head turned out to be an incorrect formula? Are there easy ways to remember the six new identities presented in this section? Group members should address this question, considering one identity at a time. For each formula, list ways to make it casier to remember.

Use this information to solve. When throwing an object, the distance achieved depends on its initial velocity, \(v_{0}\) and the angle above the horizontal at which the object is thrown, \(\theta\) The distance, \(d\), in feet, that describes the range covered is given by $$d=\frac{v_{0}^{2}}{16} \sin \theta \cos \theta$$ where \(v_{0}\) is measured in feet per second. You and your friend are throwing a baseball back and forth. If you throw the ball with an initial velocity of \(v_{0}=90\) feet per second, at what angle of elevation, \(\theta,\) to the nearest degree, should you direct your throw so that it can be easily caught by your friend located 170 feet away?

Describe the difference between verifying a trigonometric identity and solving a trigonometric equation.

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. $$\sin x-\sin x \cos ^{2} x=\sin ^{3} x$$

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. $$3 \tan ^{2} x-\tan x-2=0$$
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