Find the principal value of ^ -1 (-1). - Vidyadip

Question

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

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Answer

The given inverse trignometric function is $\tan ^ { - 1 } ( - 1 )$
Now, $\tan ^ { - 1 } ( - 1 ) = \tan ^ { - 1 } \left[ - \tan (\frac { \pi } { 4 } \right)]$ $\left[ \because \tan \frac { \pi } { 4 } = 1 \right]$
$= \tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 4 } \right) \right]$ $[ \because - \tan \theta = \tan ( - \theta ) ]$
$= - \frac { \pi } { 4 } \left[ \because \tan ^ { - 1 } ( \tan \theta ) = \theta ; \forall \theta \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) \right]$
which is the required principal value.

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