Question

The set of values of x satisfying $\left[ta{n}^{-1}x\right]+\left[co{t}^{-1}x\right]=2$ where [.] is greatest integer function is

• A. (cot 3, cot 2)
• B. (cot 3, -cot 1)
• C. (cot 3, 0)
• D. $\varphi$

Answer 1

The correct option is D

$\varphi$

$\left[co{t}^{-1}x\right]=0,1,2,3,\left[ta{n}^{-1}x\right]=-2,-1,0,1$

$\text{Case}\left(i\right)\left[co{t}^{-1}x\right]=\left[ta{n}^{-1}x\right]=1$

$x\in (cot2,cot1],\&x\in [tan1,\infty )$

$\therefore x\in \varphi ascot1<tan1$

$\left(ii\right)\left[co{t}^{-1}x\right]=3,\left[ta{n}^{-1}x\right]=-1$

$x\in (-\infty ,cot3],x\in [-tan1,0]$

$\therefore x\in \varphi ascot3<-tan1$

$\left(iii\right)\left[co{t}^{-1}x\right]=2,\left[ta{n}^{-1}x\right]=0$

$x\in (cot3,cot2],x\in [0,tan1)$

$x\in \varphi ascot2<0$

So no solution.