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Chapter 2: Problem 12

\(\frac{1}{\tan \beta}+\tan \beta=\frac{\sec ^{2} \beta}{\tan \beta}\)

Short Answer

Expert verified
The given equation is not correct.

Step by step solution

01

Simplify the right side of the equation

Start the calculation by simplifying the right side of the equation, which is \({\sec ^{2} \beta}/{\tan \beta}\). Replace \(\sec \beta\) with \(1/\cos \beta\) and \(\tan \beta\) with \(\sin \beta/\cos \beta\). Hence, the right side becomes \({(1/{\cos ^2 \beta})}/{(\sin \beta/\cos \beta)}\).
02

Further operations on the right side

Proceeding from Step 1, simplify further by multiplying by the reciprocal of the denominator, thereby cancelling out some terms. So, the expression becomes \((1/{\cos ^2 \beta}) \cdot (\cos \beta/\sin \beta) = \frac{\cos \beta}{\sin \beta \cos \beta}\). This can be further simplified to \(1/\sin \beta\).
03

Express \(1/\sin \beta\) in terms of tangent

Finally, replace \(1/\sin \beta\) with \(\cot \beta\), which gives the right side of the equation as \(cot \beta\).
04

Simplify the left side of the equation

Now, note that the left side of the equation is \(\frac{1}{\tan \beta} + \tan \beta\). Write \(\frac{1}{\tan \beta}\) as \(\cot \beta\). So, the left side simplifies to \(\cot \beta + \tan \beta\).
05

Comparing both sides of the equation

The left-hand side of the equation is now \(\cot \beta + \tan \beta\), and the right-hand side of the equation is \(\cot \beta\). Since \(\cot \beta \neq \cot \beta + \tan \beta\), the given equation is not valid.

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In Exercises 55-64, verify the identity. $$ \sin \left(\frac{\pi}{6}+x\right)=\frac{1}{2}(\cos x+\sqrt{3} \sin x) $$

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