Simplify tan(arcsin(u))

Problem

tan(arcsin(u))

Solution

1. Identify the inner function as an angle θ where θ=arcsin(u)

2. Rewrite the relationship using the definition of the inverse sine function, which implies sin(θ)=u

3. Construct a right triangle where the side opposite to θ is u and the hypotenuse is 1 satisfying sin(θ)=opposite/hypotenuse=u/1

4. Apply the Pythagorean theorem to find the adjacent side a where a2+u2=1

5. Solve for the adjacent side, resulting in a=√(,1−u2)

6. Determine the tangent of the angle θ using the ratio tan(θ)=opposite/adjacent

7. Substitute the side lengths into the ratio to get tan(θ)=u/√(,1−u2)

Final Answer

tan(arcsin(u))=u/√(,1−u2)

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