Mathematics - Trigonometric Equation Question with Solution | TestHub

Exactly one $x$ in $(0,\pi )$ needs $\sin x=1$ as one root and the other root $\sin x=p$ with $p\ge 1$ or $p\le 0$. So ${\sin }^{2}x-\alpha \sin x+\beta =(\sin x-1)(\sin x-p)$, giving $\alpha =p+1$ and $\beta =p$. Since $p\ge 1$ or $p\le 0,\alpha \ge 2$ or $\alpha \le 1$, so $\alpha$ never lies in (1, 2). Only option D can hold.