Simplify arccsc(-2/( square root of 2))

Problem

arccsc(−2/√(,2))

Solution

1. Simplify the argument of the function by rationalizing the fraction.

2/√(,2)=(2√(,2))/2=√(,2)

2. Rewrite the expression using the simplified argument.

arccsc(−√(,2))

3. Apply the identity for inverse cosecant with a negative argument, which is arccsc(−x)=−arccsc(x)

−arccsc(√(,2))

4. Relate the inverse cosecant to the inverse sine using the identity arccsc(x)=arcsin(1/x)

−arcsin(1/√(,2))

5. Evaluate the inverse sine by finding the angle θ in the interval [−π/2,π/2] such that sin(θ)=1/√(,2)

arcsin(1/√(,2))=π/4

6. Combine the results to find the final value.

−π/4

Final Answer

arccsc(−2/√(,2))=−π/4

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