Question

If $2\le a\le 3$, then the value of ${\cos }^{-1}\cos \left[a\right]+{\csc }^{-1}\csc \left[a\right]+{\cot }^{-1}\cot \left[a\right]$. (where $[.]$ denotes greatest integer less than equal to $x$) is equal to

• A. $2-\pi$
• B. $3-\pi$
• C. $\pi$
• D. $6$

Answer 1

The correct option is A $2-\pi$

$\left[a\right]$ possibly can have $2$ values $2,3$

For $\left[a\right]=2,i.e.,2\le a<3$

$co{s}^{-1}cos2+cs{c}^{-1}csc2+co{t}^{-1}cot2$

$co{s}^{-1}cos(\pi -(\pi -2\left)\right)+cs{c}^{-1}csc(\pi -(\pi -2\left)\right)+co{t}^{-1}cot(\pi -(\pi -2\left)\right)$

$2-\pi +\pi -2-\pi +2=2-\pi$

For $a=3$

${\text{cos}}^{-1}cos3+{\text{csc}}^{-1}\csc 3+{\text{cot}}^{-1}\cot 3$

${\text{cos}}^{-1}cos(\pi -(\pi -3\left)\right)+{\text{csc}}^{-1}\csc (\pi -(\pi -3\left)\right)+{\text{cot}}^{-1}\cot (\pi -(\pi -3\left)\right)$

$3-\pi +\pi -3-\pi +3=3-\pi$