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Chapter 5: Problem 22

Verify each identity. $$\frac{\cot ^{2} t}{\csc t}=\csc t-\sin t$$

Short Answer

Expert verified
The short answer to the exercise is that the presented equation is indeed valid as both sides simplify to the same expression, justifying the identity.

Step by step solution

01

Rewrite in terms of sines and cosines

Write the given equation in terms of sines and cosines. Because \(\cot t=\frac{\cos t}{\sin t}\) and \(\csc t=\frac{1}{\sin t}\), it is possible to rewrite the left-hand side of the equation as \(\frac{(\frac{\cos t}{\sin t})^{2}}{(\frac{1}{\sin t})}\). Similarly \(\csc t - \sin t\) can be rewritten as \(\frac{1}{\sin t} - \sin t \).
02

Simplify the Equation

Further simplifying the equation results in \(\frac{\cos^{2} t}{\sin^{2} t} = \frac{1}{\sin t} - \sin t\). This can be manipulated to form \(\frac{\cos^{2} t}{\sin^{2} t} = \frac{1 - \sin^{2} t}{\sin t} \).
03

Use Pythagorean Identity

Using the Pythagorean identity, which states that \(\cos^{2}t + \sin^{2}t = 1\), allows for simplification of the equation. Substitute \(1 - \sin^{2} t\) for \(\cos^{2} t\) in the equation to get \(\frac{1 - \sin^{2} t}{\sin^{2} t} = \frac{1 - \sin^{2} t}{\sin t}\).

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