If $ [cot^{-1}x]  +[cos^{-1}x]=0$ where x is a non-negative real number and [.] denotes the greatest integer function, then complete set of values of x, is

(cot 1, 1]

We have,

$cot^{-1} x ∈[0, \pi ]$ and $ cos^{-1} x ∈[0, \pi]$

$∴ [cot^{-1}x] > 0 $ and $[cos^{-1}x] ≥ 0$

Now, $[cot^{-1}x] +[cos^{-1}x]=0$

$⇒ [cot^{-1}x]= 0 = [cos^{-1}x]$

$⇒ 0 < cot^{-1}x < 1$ and $ 0 ≤ cos^{-1}x < 1 $

$⇒ cot 1 < x < ∞ $ and $ cos 1 < x ≤1$

$⇒ x ∈ (cot 1, ∞)$ and $ x ∈ (cos 1, 1]⇒ x ∈(cot 1, 1].$