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Chapter 4: Problem 65

Write an algebraic expression that is equivalent to the given expression. $$\cot (\arctan x)$$

Short Answer

Expert verified
The algebraic expression equivalent to \( \cot (\arctan x) \) is \( \frac{1}{x} \)

Step by step solution

01

Recognize the inverse trigonometric function

In the expression \( \cot (\arctan x) \), \(\arctan x\) is an inverse trigonometric function that returns an angle whose tangent is \(x\). So, the problem reduces to finding what is the cotangent of an angle whose tangent is \(x\).
02

Express in form of cotangent

We know that cotangent is the reciprocal of tangent. So, cotangent of an angle is the ratio of the adjacent side to the opposite side in a right triangle formed by the angle. Therefore, we can express it in terms of \(x\) as \( \cot (\arctan x) = \frac{1}{\tan (\arctan x)} \)
03

Simplify the expression

Since the tangent of the arc tangent of \(x\) is simply \(x\) (because \(\tan (\arctan x) = x\)), we can replace \(\tan (\arctan x)\) in the above expression. Hence, \( \cot (\arctan x) = \frac{1}{x} \)

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