if [sin-1x]+ [cos-1x]=0 where x is an non negativereal number and [.] denotes the greatest integer function , then complete set of values of x is -

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \left[{\sin }^{-1}\left(\mathrm{x}\right)\right]+\left[{\cos }^{-1}\left(\mathrm{x}\right)\right]=0 \\ \mathrm{Now} \mathrm{domain} \mathrm{of} {\sin }^{-1}\mathrm{x} \mathrm{and} {\cos }^{-1}\mathrm{x} \mathrm{is} \left[-1,1\right] \mathrm{but} \mathrm{x} \mathrm{is} \mathrm{non}-\mathrm{negative}, \mathrm{hence} \\ \mathrm{x}\in \left[0,1\right] \\ \mathrm{Now} \mathrm{x}\in \left[0,1\right] {\sin }^{-1}\mathrm{x}\in \left[0,\frac{\mathrm{\pi }}{2}\right] \mathrm{and} {\cos }^{-1}\mathrm{x}\in \left[\frac{\mathrm{\pi }}{2},0\right] \\ \mathrm{So} \left[{\sin }^{-1}\left(\mathrm{x}\right)\right]+\left[{\cos }^{-1}\left(\mathrm{x}\right)\right]=0 \mathrm{when} \\ \left[{\sin }^{-1}\left(\mathrm{x}\right)\right]=0 \mathrm{and} \left[{\cos }^{-1}\left(\mathrm{x}\right)\right]=0 \\ \left[{\sin }^{-1}\left(\mathrm{x}\right)\right]=0 \mathrm{when} \mathrm{x}\in [0,\sin 1) \left[\mathrm{As} {\sin }^{-1}\left(\sin 1\right)=1\right] \\ \mathrm{Similarly} \\ \left[{\cos }^{-1}\left(\mathrm{x}\right)\right]=0 \mathrm{when} \mathrm{x}(\cos 1,1] \left[\mathrm{As} {\cos }^{-1}\left(\cos 1\right)=1\right] \\ \mathrm{Taking} \mathrm{intersection} \mathrm{o} \mathrm{ftwo} \mathrm{conditions} \mathrm{we} \mathrm{get} \\ \mathbf{x}\mathbf{\in }\left(\mathbf{cos}\mathbf{1}\mathbf{,}\mathbf{sin}\mathbf{1}\right) \end{gathered}$$

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