Find the principal values of the following questions:

$t a n^{- 1} (- 1)$

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Solution

Let $t a n^{- 1} (- 1) = θ \Rightarrow t a n θ = - 1$
We know that the range of principal value of $t a n^{- 1} θ i s (- \frac{π}{2}, \frac{π}{2})$ .
$∴ t a n θ = - 1 = - t a n \frac{π}{4} = t a n (- \frac{π}{4}) (∵ t a n (- θ) = - t a n θ)$
$\Rightarrow θ = - \frac{π}{4}$ where $θ ϵ (- \frac{π}{2}, \frac{π}{2}) ∴ t a n^{- 1} (- 1) = - \frac{π}{4}$
Hence, the principal value of $t a n^{- 1} (- 1)$ is $- \frac{π}{4}$

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