Question

Find the value of
$\left[\cos 1-{\cos }^{-1}1\right]-\left[\sin 1-{\sin }^{-1}1\right]+\left[\tan 1-{\tan }^{-1}1\right]-\left[\cot 1-{\cot }^{-1}1\right]+\left[\sec 1-{\sec }^{-1}1\right]-\left[\mathrm{cosec}1-{\mathrm{cosec}}^{-1}1\right]$
where $[x]$ denotes greatest integer less than or equal to $x$.

Answer 1

Answer: 4

As $\cos 1-{\cos }^{-1}1=\cos 1-0\in (0,1)\Rightarrow \left[\cos 1-{\cos }^{-1}1\right]=0$
$\sin 1-{\sin }^{-1}1=\sin 1-\frac{\pi }{2}\in (-1,0)\Rightarrow \left[\sin 1-{\sin }^{-1}1\right]=-1$
$\tan 1-{\tan }^{-1}1=\tan 1-\frac{\pi }{4}\approx 1.73-0.78>0\text{ and }\in (0,1)\Rightarrow \left[\tan 1-{\tan }^{-1}1\right]=0\left\{\tan 1\approx \tan \frac{\pi }{3}\right\}$
$\cot 1-{\cot }^{-1}1=\cot 1-\frac{\pi }{4}\approx 0.57-0.78<0\text{ and }\in (-1,0)\Rightarrow \left[\cot 1-{\cot }^{-1}1\right]=-1$
$\sec 1-{\sec }^{-1}1=\sec 1-0<\sec \frac{\pi }{3}=2.\text{ As }\sec 1\in (1,2)\Rightarrow \left[\sec 1-{\sec }^{-1}1\right]=1$
$\text{ and }\mathrm{cosec}1-{\mathrm{cosec}}^{-1}1=\mathrm{cosec}1-\frac{\pi }{2}$
As $\mathrm{cosec}\frac{\pi }{3}<\mathrm{cosec}1<\mathrm{cosec}\frac{\pi }{4}<\frac{\pi }{2}\Rightarrow \mathrm{cosec}1\in \left(\frac{2}{\sqrt{3}},\sqrt{2}\right)$
$\therefore \mathrm{cosec}1-\frac{\pi }{2}\in (-1,0)\Rightarrow \left[\mathrm{cosec}1-{\mathrm{cosec}}^{-1}1\right]=-1$
Hence the value of given exprersion $=0-(-1)+0-(-1)+1-(-1)=1+1+1+1=4$