Simplify cot(arcsin(x))

Problem

cot(arcsin(x))

Solution

1. Identify the inner function and set it equal to an angle θ

θ=arcsin(x)

2. Rewrite the inverse trigonometric relationship as a trigonometric equation.

sin(θ)=x

3. Interpret the sine ratio in terms of a right triangle where sin(θ)=opposite/hypotenuse

opposite=x

hypotenuse=1

4. Apply the Pythagorean theorem to find the adjacent side a

a2+x2=1

a2=1−x2

a=√(,1−x2)

5. Express the cotangent function in terms of the sides of the triangle.

cot(θ)=adjacent/opposite

6. Substitute the side lengths into the cotangent expression.

cot(θ)=√(,1−x2)/x

Final Answer

cot(arcsin(x))=√(,1−x2)/x

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