Question

Complete solution set $\left[{\cot }^{-1}x\right]+2\left[{\tan }^{-1}x\right]=0$, where $[.]$ denotes the greatest integer function, is

• A. $(0,\cot 1)$
• B. $(0,\tan 1)$
• C. $(\tan 1,\infty )$
• D. $(\cot 1,\tan 1)$

Answer 1

The correct option is D $(\cot 1,\tan 1)$

$\left[{\cot }^{-1}x\right]+2\left[{\tan }^{-1}x\right]=0\Rightarrow \left[{\cot }^{-1}x\right]=0,\left[{\tan }^{-1}x\right]=0$

or $\left[{\cot }^{-1}x\right]=2,\left[{\tan }^{-1}x\right]=-1$

Now $\left[{\cot }^{-1}x\right]=0\Rightarrow x\in (\cot 1,\infty )$

$\left[{\tan }^{-1}x\right]=0\Rightarrow x\in (0,\tan 1)$
Therefore, for
$\left[{\cot }^{-1}x\right]=\left[{\tan }^{-1}x\right]=0,x\in (\cot 1,\tan 1)$

$\left[{\cot }^{-1}x\right]=2\Rightarrow x\in (\cot 3,\cot 2]$

$\left[{\tan }^{-1}x\right]=-1\Rightarrow x\in [-\tan 1,0]\Rightarrow \text{No such exist.}$

Thus, the solution set is $(\cot 1,\tan 1).$