Find the principal value of \cosec^{-1} (-\sqrt{2})

Anonymous
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Step 1: Calculate the values of the inverse function.
GIVEN: \cosec^{-1}(-\sqrt{2})
Let \cosec^{-1}(-\sqrt{2}) = y
Where \frac{-\pi}{2} < y < \frac{-\pi}{2}
\cosec (y) = (-\sqrt{2})
\cosec (y) = \cosec (\frac{-\pi}{4}) \because \cosec (\frac{-\pi}{4}) =-\sqrt{2}
\cosec y = - \cosec (\frac{\pi}{4}) \because \cosec (- \theta) =- \cosec \theta
y = - \frac{\pi}{4}
\therefore \text{The principal value of the} \cosec (\sqrt{2}) \text{ is } \frac{-\pi}{4}