Simplify cos(arcsin(u))

Problem

cos(arcsin(u))

Solution

1. Identify the relationship between the trigonometric functions by letting θ=arcsin(u)

2. Rewrite the expression in terms of θ as cos(θ) where sin(θ)=u

3. Use the Pythagorean identity sin2(θ)+cos2(θ)=1 to relate the two functions.

4. Solve for cos(θ) by substituting u for sin(θ) resulting in u2+cos2(θ)=1

5. Isolate the cosine term to get cos2(θ)=1−u2

6. Take the square root, noting that the range of arcsin(u) is [−π/2,π/2] where cosine is non-negative.

Final Answer

cos(arcsin(u))=√(,1−u2)

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