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)

$\frac{\pi}{2} \,\text{for}\, \text{all}\, x$

Answer 2

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, $\tan^{-1} x + \cot^{-1} x = p + \frac{\pi}{2} - p$ $= \frac{\pi}{2} , \forall x \in R$