Question

Let \(m\) and \(n\) be non–negative integers such that for \[ x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right),\qquad \tan x+\sin x=m,\quad \tan x-\sin x=n. \] Then the possible ordered pair \((m,n)\) is:

• A. \((2,1)\) but not \((3,4)\)
• B. \((3,4)\) but not \((2,1)\)
• C. both \((2,1)\) and \((3,4)\)
• D. neither \((2,1)\) nor \((3,4)\)

Hint:

When trigonometric expressions are given as sums and differences, always reduce them to \(\sin x\) and \(\tan x\), then use \(\sin^2 x+\cos^2 x=1\) to check consistency.

Answer 1

The Correct Option is D