Simplify csc(arccos(x))

Problem

csc(arccos(x))

Solution

1. Define an angle θ such that θ=arccos(x) This implies that cos(θ)=x where 0≤θ≤π

2. Represent the relationship using a right triangle where the adjacent side is x and the hypotenuse is 1

3. Find the opposite side of the triangle using the Pythagorean theorem a2+b2=c2

x2+opposite2=1

opposite=√(,1−x2)

4. Identify the cosecant of the angle θ which is defined as the ratio of the hypotenuse to the opposite side.

csc(θ)=1/√(,1−x2)

Final Answer

csc(arccos(x))=1/√(,1−x2)

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