Question

If $2\le a<3$, then the value of ${\cos }^{-1}\cos \left[a\right]+{\text{cosec}}^{-1}\text{cosec}\left[a\right]+{\cot }^{-1}\cot \left[a\right]$. where [.] denotes greater integer less then equal to $x$) is equal to:

• A. $2-\pi$
• B. $2+\pi$
• C. $\pi$
• D. $6$

Answer 1

Answer: (D)

$6$

$[.]\to$ greatest Integer function

$\therefore \left[a\right]=2$ for $2\le a<3$

Also, ${\sin }^{-1}\left(\sin x\right)=x,{\cos }^{-1}\left(\cos x\right)=x$

${\text{cosec}}^{-1}\left(\text{cosec}x\right)=x,{\cot }^{-1}\left(\cot x\right)=x$

Now,

${\cos }^{-1}\cos \left[a\right]+{\text{cosec}}^{-1}\text{cosec}\left[a\right]+{\cot }^{-1}\cot \left[a\right]$

$={\cos }^{-1}\cos 2+{\text{cosec}}^{-1}\text{cosec}2+{\cot }^{-1}\cot 2$

$=2+2+2=6$