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Chapter 4: Problem 46

In Exercises \(37-46,\) use trigonometric identities to transform the left side of the equation into the right side \((0<\theta<\pi / 2) .\) $$\frac{\tan \beta+\cot \beta}{\tan \beta}=\csc ^{2} \beta$$

Short Answer

Expert verified
After proper application of the trigonometric identities, the left-hand side \(\frac{\tan\ \beta\ +\ \cot\ \beta}{\tan\ \beta}\) of the equation simplifies to \(\csc^2\ \beta\) which is same as the right-hand side.

Step by step solution

01

Break Down the Expression

Our task is to simplify the left side expression into the right one. The left side of the expression can be broken down as: \(\frac{\tan\ \beta\ +\ 1/\tan\ \beta}{\tan\ \beta}\)
02

Simplify The Expression

After rewriting the tangent terms and cotangent terms, the expression is simplified to: \(\tan\ \beta+\frac{1}{\tan\ \beta}\) divided by \(\tan\ \beta\). When divided by \(\tan\ \beta\), it simplifies to \(1 + \frac{1}{\tan^2\ \beta}\)
03

Replace 1/tan with cot

Now, we can replace \(1/\tan\ \beta\) with \(\cot\ \beta\) which gives \(1 + \cot^2\ \beta\) going by the property \(\cot^2\ \beta = \csc^2\ \beta - 1\). So, \(1 + \cot^2\ \beta\) becomes \(\csc^2\ \beta\)
04

Compare the Simplified Expression with RHS

Now, the left hand side becomes \(\csc^2\ \beta\) which is same as the right hand side.

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