Question

If $\left[co{t}^{-1}x\right]+\left[co{s}^{-1}x\right]=0$, where $[\cdot ]$ denotes the greatest integer function, then the complete set of values of $x$ is

• A. $(cos1,1]$
• B. $(cos1,-cos1)$
• C. $(cot1,1]$
• D. $noneofthese$

Answer 1

Answer: (D)

$noneofthese$

Re-calling the domain & range of inverse cosine & inverse cotangent function
$y=co{s}^{-1}x$ , Domain $[-1,1]$ Range $[0,\pi ]$
$y=co{t}^{-1}x$ , Domain $(-\infty ,\infty )$ Range $(0,\pi )$
As we can see range of above two functions is always positive.
So if we are expecting their greatest integer sums to be zero then it can happen if and only if both the greatest integer functions are individually zero.
Concentrating on inverse cosine function we can see, its greatest integer function will have zero values in the range $(cos1,1]$.
We can check in this range inverse cotangent function will also be zero.
Hence this is our required range of values of x.