Simplify tan(arccos(u))

Problem

tan(arccos(u))

Solution

1. Define an angle θ such that θ=arccos(u) This implies that cos(θ)=u where 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, y

u2+y2=1

4. Solve for the opposite side y

y=√(,1−u2)

5. Use the definition of the tangent function, which is the ratio of the opposite side to the adjacent side.

tan(θ)=opposite/adjacent

6. Substitute the values found from the triangle into the tangent ratio.

tan(θ)=√(,1−u2)/u

Final Answer

tan(arccos(u))=√(,1−u2)/u

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