Simplify csc(arccos(u))

Problem

csc(arccos(u))

Solution

1. Identify the inner function and set it equal to an angle θ Let θ=arccos(u) which implies cos(θ)=u for 0≤θ≤π

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

3. Apply the Pythagorean theorem to find the length of the opposite side. If a2+b2=c2 then opposite2+u2=1

4. Solve for the opposite side, which results in √(,1−u2)

5. Evaluate the outer function csc(θ) using the triangle. Since csc(θ)=hypotenuse/opposite substitute the known values.

6. Substitute the expression for the opposite side into the ratio to get 1/√(,1−u2)

Final Answer

csc(arccos(u))=1/√(,1−u2)

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