Question

If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.

Answer 1

If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then – 2π < y < 2π.

Explanation:

y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))`

⇒ y = `2 tan^-1x + 2 tan^-1x`1

⇒ y = `4 tan^-1x`  ......`[because sin^1 ((x)/(1 +x^2)) = 2tan^-1x]`

Now `(-pi)/2 < tan^-1x < pi/2`

⇒ `-4 xx pi/2 < 4 tan^-1x < 4 xx pi/2`

⇒  – 2π < y < 2π.