Solve If $[\cos^{-1}x] + [\cot ^{-1}x] = 0$, where [.] denotes the greate integer function.

$(\cot 1, 1]$

We have $[\cos^{-1} x] ≥0\,∀\,x∈ [-1, 1]$

and $[\cot^{-1} x] ≥0\,∀\,x∈R$

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

only if $[\cos^{-1}x] = [\cot ^{-1}x] = 0$

$[\cos^{-1}x] = 0⇒x∈ (\cos 1, 1]$

$[\cot^{-1}x] = 0⇒x∈ (\cot 1, ∞)$

$∴x∈ (\cot 1, 1]$