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. Recall that the range of the arcsine function is [−π/2,π/2]

3. Identify the angle θ such that sin(θ)=−√(,2)/2 within that range.

θ=−π/4

4. Substitute the angle back into the original expression to find sec(−π/4)

sec(−π/4)

5. Apply the reciprocal identity sec(x)=1/cos(x)

sec(−π/4)=1/cos(−π/4)

6. Use the even function property cos(−x)=cos(x) and the known value cos(π/4)=√(,2)/2

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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