Solve If $[\cos^{-1}x] + [\cot ^{-1}x] = 0$, where [.] denotes the greate integer function.
Answer & explanation
Correct answer: option 2
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 $[\cos^{-1}x] = [\cot ^{-1}x] = 0$
$[\cos^{-1}x] = 0⇒x∈ (\cos 1, 1]$
$[\cot^{-1}x] = 0⇒x∈ (\cot 1, ∞)$
$∴x∈ (\cot 1, 1]$