Answer:
sec(-π +ln(1+√2)i) = -√2/2 . . . . if complex θ is allowed
Explanation:
For real values of θ, each of the trig functions has a specified range. The value -√2/2 is outside the range of the secant function, so sec(θ) = -√2/2 cannot exist for real values of θ.
In geometrical terms, the secant of an angle is the ratio of the hypotenuse to the near side of that angle in a right triangle. The hypotenuse can never be shorter than the side length, so |sec(θ)| = √2/2 is geometrical nonsense.
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Additional comment
The ranges of all trig functions include all complex numbers when complex values of their arguments are allowed. For the case at hand, the value of θ for sec(θ) = -√2/2 is ...
θ ≈ -π +ln(1+√2)i or π +ln(√2 -1)i
This value can be found by using Euler's formula to solve the given equation.
Some calculators are equipped to provide the numerical value of the complex arcsecant of -√2/2.