Question

For \(x\in\mathbb{R}\), compute \(\csc\!\big(\tan^{-1}x+\cot^{-1}x\big)\).

• A. \(0\)
• B. \(1\)
• C. \(\dfrac{2}{\sqrt3}\)
• D. \(2\)

Hint:

Remember $\arctan x+\operatorname{arccot}x=\dfrac{\pi}{2}$ (principal values).

Answer 1

The Correct Option is B

With the standard principal ranges used in school exams, \(\tan^{-1}x+\cot^{-1}x=\dfrac{\pi}{2}\). Therefore \[ \csc\!\left(\tan^{-1}x+\cot^{-1}x\right)=\csc\!\left(\frac{\pi}{2}\right)=1. \]