Question

What is the value of $\tan^{-1} x + \cot^{-1} x $ ?

• A. $\frac{\pi}{2}\, \text{for} \,x > 0 \text{and} - \,\frac{\pi}{2} \,\text{for}\, x < 0 $
• B. $\frac{\pi}{2} \,\text{for}\, \text{all}\, x $
• C. $ - \frac{\pi}{2} \,\text{for}\, \text{all}\, x $
• D. $\frac{\pi}{2}\, \text{for} \,\text{integral} \,x $

Answer 1

Answer: (B)

Let $\tan^{-1} x = p $ so, $\tan p = x$
$ \tan p = \cot \left(\frac{\pi}{2} - p \right) $
$\Rightarrow x = \cot \left(\frac{\pi}{2} - p \right) \Rightarrow \cot^{-1} x = \frac{\pi}{2} - p$
So, $ \tan^{-1} x + \cot^{-1} x = p + \frac{\pi}{2} - p$
$ = \frac{\pi}{2} , \forall x \in R $