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Chapter 4: Problem 72

Write an algebraic expression that is equivalent to the given expression. $$\cot \left(\arctan \frac{1}{x}\right)$$

Short Answer

Expert verified
The equivalent algebraic expression to the given expression is \(x\).

Step by step solution

01

Understanding the trigonometric identities

In this case, the arctangent of a number \(z\), often written as \(\arctan (z)\), is the angle whose tangent is \(z\). Similarly, the cotangent of an angle is the reciprocal of the tangent of the angle.
02

Substitute the arctangent expression

The given expression can be rewritten as \(\cot (\theta)\), where \(\theta = \arctan \frac{1}{x}\). This means that \(\tan (\theta) = \frac{1}{x}\).
03

Convert cotangent to reciprocal of tangent

As the cotangent of \(\theta\) is the reciprocal of the tangent of \(\theta\), the expression \(\cot (\theta)\) can be rewritten as \(\frac{1}{\tan (\theta)}\). Substituting the expression of \(\tan (\theta)\), we get \(\frac{1}{\frac{1}{x}} = x\).

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