Simplify sec(arcsin(( square root of 2)/2))

Problem

sec(arcsin(√(,2)/2))

Solution

1. Evaluate the inner inverse trigonometric function arcsin(√(,2)/2)

2. Identify the angle θ in the interval [−π/2,π/2] such that sin(θ)=√(,2)/2

3. Determine that θ=π/4 because sin(π/4)=√(,2)/2

4. Substitute the angle back into the outer function to get sec(π/4)

5. Apply the definition of the secant function, which is the reciprocal of the cosine function: sec(π/4)=1/cos(π/4)

6. Calculate the value using cos(π/4)=√(,2)/2

7. Simplify the resulting fraction 1/√(,2)/2=2/√(,2)=√(,2)

Final Answer

sec(arcsin(√(,2)/2))=√(,2)

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