Find the principal value of ${tan}^{- 1} (- 1) .$

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Solution

Let ${tan}^{- 1} (- 1) = y$ .

Then, $tan y = - 1 = - tan (\frac{π}{4}) = tan (- \frac{π}{4})$ .

We know that the range of the principal value of ${tan}^{- 1} x$ is $(- \frac{π}{2}, \frac{π}{2})$ and $tan (- \frac{π}{4}) = - 1$ .

Therefore, the principal value of ${tan}^{- 1} (- 1) x$ is $- \frac{π}{4}$

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