Consider equation $(\mathrm{x}-\sin \mathrm{\alpha })(\mathrm{x}-\cos \mathrm{\alpha })-2=0$. Which of the following is/are true?

Let $\mathrm{f}\left(\mathrm{x}\right)=(\mathrm{x}-\sin \mathrm{\alpha })(\mathrm{x}-\cos \mathrm{\alpha })-2$.

Then, $\mathrm{f}(\sin \mathrm{\alpha })=-2<0$ and $\mathrm{f}(\cos \mathrm{\alpha })=-2<0$.

So, $\sin \mathrm{\alpha }\text{ and }\cos \mathrm{\alpha }$ lie between the roots.

For $0<\mathrm{\alpha }<\frac{\mathrm{\pi }}{4},\sin \mathrm{\alpha }<\cos \mathrm{\alpha }$

Therefore, equation f (x) = 0 has one root in $(-\mathrm{\infty }, \sin \mathrm{\alpha })$ and other in $(\cos \mathrm{\alpha }, \mathrm{\infty })$.

Also, for $\frac{\mathrm{\pi }}{4}<\mathrm{\alpha }<\frac{\mathrm{\pi }}{2}$, $\cos \mathrm{\alpha }<\sin \mathrm{\alpha }$

Therefore, equation f(x) = 0 has one root in $(-\mathrm{\infty }, \cos \mathrm{\alpha })$ and other in $(\sin \mathrm{\alpha }, \mathrm{\infty })$.