Find the principal values of the following questions:

$c o s c e^{- 1} (- \sqrt{2})$

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Solution

Let $c o s c e^{- 1} (- \sqrt{2})$
We know that the range of principal value of $c o s e c^{- 1} θ$ is $[- \frac{π}{2}, \frac{π}{2}] - {0},$
$∴ c o s e c θ = - \sqrt{2} = - c o s e c \frac{π}{4} = c o s e c (- \frac{π}{4}) (∵ c o s e c (- θ) = - c o s e c θ)$
$\Rightarrow θ = - \frac{π}{4}$ , where $θ ε [- \frac{π}{2}, \frac{π}{2}] - {0} \Rightarrow c o s e c^{- 1} (- \sqrt{2}) = - \frac{π}{4}$
Hence, the principal value of $c o s e c^{- 1} (- \sqrt{2}) i s - \frac{π}{4}$

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