Simplify cot(arcsin(u))

Problem

cot(arcsin(u))

Solution

1. Identify the inner function and let θ=arcsin(u) This implies that sin(θ)=u where θ is in the interval [−π/2,π/2]

2. Represent the trigonometric relationship using a right triangle where the side opposite to angle θ is u and the hypotenuse is 1

3. Apply the Pythagorean theorem to find the adjacent side length a

a2+u2=1

a=√(,1−u2)

4. Use the definition of the cotangent function, which is the ratio of the adjacent side to the opposite side.

cot(θ)=adjacent/opposite

5. Substitute the side lengths into the ratio to find the simplified expression.

cot(θ)=√(,1−u2)/u

Final Answer

cot(arcsin(u))=√(,1−u2)/u

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