Simplify sec(arcsin(x-1))

Problem

sec(arcsin(x−1))

Solution

1. Identify the inner expression as an angle θ such that θ=arcsin(x−1)

2. Relate the sine of the angle to the expression by the definition of the inverse sine function, which gives sin(θ)=(x−1)/1

3. Construct a right triangle where the side opposite to θ is x−1 and the hypotenuse is 1

4. Apply the Pythagorean theorem to find the adjacent side a of the triangle using a2+(x−1)2=1

5. Solve for the adjacent side a=√(,1−(x−1)2)

6. Expand the expression under the radical to simplify the adjacent side to a=√(,1−(x2−2*x+1)) which results in a=√(,2*x−x2)

7. Determine the secant of θ using the ratio of the hypotenuse over the adjacent side, sec(θ)=1/a

8. Substitute the simplified adjacent side back into the ratio to find the final expression.

Final Answer

sec(arcsin(x−1))=1/√(,2*x−x2)

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