Simplify cot(arccos(u))

Problem

cot(arccos(u))

Solution

1. Identify the inner function and set it equal to an angle θ such that θ=arccos(u)

2. Rewrite the relationship using the definition of the inverse cosine function, which gives cos(θ)=u for 0≤θ≤π

3. Represent the angle θ in a right triangle where the adjacent side is u and the hypotenuse is 1 since cos(θ)=adjacent/hypotenuse=u/1

4. Determine the length of the opposite side using the Pythagorean theorem: a^2+b^2=c^2

5. Solve for the opposite side: opposite=√(,1^2−u^2)=√(,1−u^2)

6. Apply the definition of the cotangent function, which is cot(θ)=adjacent/opposite

7. Substitute the side lengths into the cotangent ratio to get u/√(,1−u^2)

Final Answer

cot(arccos(u))=u/√(,1−u^2)

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