Simplify sec(arcsin(u))

Problem

sec(arcsin(u))

Solution

1. Identify the inner function and set it equal to an angle θ such that θ=arcsin(u)

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

3. Interpret the sine ratio in terms of a right triangle where sin(θ)=opposite/hypotenuse Let the opposite side be u and the hypotenuse be 1

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

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

6. Determine the secant of the angle θ using the ratio sec(θ)=hypotenuse/adjacent

7. Substitute the known values into the ratio to get sec(θ)=1/√(,1−u2)

Final Answer

sec(arcsin(u))=1/√(,1−u2)

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