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