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Chapter 9: Problem 23

Simplify $$ \tan A+\frac{1}{\tan A} $$

Short Answer

Expert verified
Answer: The simplified form of the expression \(\tan A + \frac{1}{\tan A}\) is \(\csc A\).

Step by step solution

01

Find a common denominator

To simplify the given expression, first, we need to find a common denominator. In this case, the common denominator is \(\tan A\). So, rewrite the expression with the common denominator \(\tan A\): $$ \tan A + \frac{1}{\tan A} = \frac{\tan^2 A + 1}{\tan A} $$
02

Use the Pythagorean Identity

We can use the Pythagorean Identity for the tangent function, which is given by $$ \tan^2 A + 1 = \sec^2 A $$ Replace the numerator of the expression with \(\sec^2 A\). $$ \frac{\tan^2 A + 1}{\tan A} = \frac{\sec^2 A}{\tan A} $$
03

Express secant in terms of cosine

We can express the secant function as the reciprocal of the cosine function, as follows: $$ \sec A = \frac{1}{\cos A} $$ Replace the \(\sec\) function in the expression with \(\frac{1}{\cos}\). $$ \frac{\sec^2 A}{\tan A} = \frac{\left(\frac{1}{\cos A}\right)^2}{\tan A} $$
04

Express tangent in terms of sine and cosine

Next, we express the tangent function as the ratio of the sine and cosine functions, as follows: $$ \tan A = \frac{\sin A}{\cos A} $$ Replace the \(\tan\) function with the quotient of the sine and cosine functions. $$ \frac{\left(\frac{1}{\cos A}\right)^2}{\frac{\sin A}{\cos A}} = \frac{\cos^{-2} A}{\frac{\sin A}{\cos A}} $$
05

Simplify the expression

Finally, simplify the expression by inverting and multiplying. $$ \frac{\cos^{-2} A}{\frac{\sin A}{\cos A}} = \frac{1}{\cos^2 A} \cdot \frac{\cos A}{\sin A} = \frac{\cos A}{\sin A \cos A} = \frac{1}{\sin A} $$ The simplified expression is \(\frac{1}{\sin A}\), which is equivalent to \(\csc A\). Therefore, the simplified expression is $$ \tan A+\frac{1}{\tan A} = \csc A $$

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Most popular questions from this chapter

Show \(\sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta\)

Convert the following angles in radians to degrees: (a) \(0.3609\) (b) \(0.4771\) (c) \(1.3692\) (d) \(\frac{\pi}{3}\) (e) \(\frac{2 \pi}{3}\) (f) \(6 \pi\) (g) \(\frac{\pi}{5}\) (h) \(\frac{3 \pi}{2}\)

Express the following angles in the form \(\alpha \pi\) radians: (a) \(90^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(120^{\circ}\) (e) \(240^{\circ}\) (f) \(72^{\circ}\) (g) \(216^{\circ}\) (h) \(135^{\circ}\) (i) \(108^{\circ}\) (j) \(270^{\circ}\)

Show \(\sin 3 A=3 \sin A \cos ^{2} A-\sin ^{3} A\)

Convert the following angles in radians to degrees: (a) \(\frac{\pi}{2}\) (b) \(\frac{\pi}{3}\) (c) \(\frac{4 \pi}{3}\) (d) \(1.25 \pi\) (e) \(1.25\) (f) \(9.6314(\mathrm{~g}) 3\)
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