Question

Write the expression in algebraic form. \(\cos (\operatorname{arccot} x)\)

Short answer

The algebraic form of \(\cos (\operatorname{arccot} x)\) is \(1 / \sqrt{1 + x^2}\)

Step-by-step solution

Trieangle Visualization

Imagine a right triangle where the adjacent side is \(1\) and the opposite side is \(x\). Here, \(\operatorname{cot}\theta=R\) relates to the ratio of adjacent over opposite, therefore the cotangent of the angle in the triangle is \(\operatorname{cot}\theta = 1/x\).

Pythagorean identity

Use the Pythagorean theorem to find the hypotenuse in terms of \(x\). The hypotenuse (\(H\)) is \(H = \sqrt{1 + x^2}\). This helps us to express everything in terms of \(x\).

Expression Simplification

Since \(\cos (\operatorname{arccot} x)\) is the cosine of an angle whose tangent is \(1/x\), and cosine is the ratio of adjacent over hypotenuse, the expression can be simplified to \(\cos (\operatorname{arccot} x) = 1 / \sqrt{1 + x^2} \).