Write the value

Question:

Write the value of $\tan ^{-1}\left(\frac{1}{x}\right)$ for $x<0$ in terms of $\cot ^{-1} x$

Solution:

$\tan ^{-1}\left(\frac{1}{x}\right)=\tan ^{-1}\left(-\frac{1}{x}\right) \quad$ for $x<0$

$=-\tan ^{-1}\left(\frac{1}{x}\right)$

$=-\cot ^{-1} x$

$=-\left(\pi-\cot ^{-1} x\right)$

$=-\pi+\cot ^{-1} x$

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