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 $x<0$
$=-\tan ^{-1}\left(\frac{1}{x}\right)$
$=-\cot ^{-1} x$
$=-\left(\pi-\cot ^{-1} x\right)$
$=-\pi+\cot ^{-1} x$