Question: Find the principal value of \begin{align} tan^{-1}\left(-1\right)\end{align}

Answer:

\begin{align} Let \;\; tan^{-1}\left(-1\right)=y. \;\;Then,\;\; tan y = -1 = -tan\left(\frac{\pi}{4}\right)= tan\left(-\frac{\pi}{4}\right)\end{align}

We know that the range of the principal value branch of tan−1 is

\begin{align} \left(-\frac{\pi}{2},\frac{\pi}{2}\right) and \;\;tan\left(-\frac{\pi}{4}\right) = - 1\end{align}

Therefore, the principal value of

\begin{align} tan^{-1}\left(- 1\right) is -\frac{\pi}{4}\end{align}

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