58th Ukrainian National Mathematical Olympiad

Let us write the equation that is necessary and sufficient for these three numbers to form geometric progression: $$\sin x\cdot \mathrm{tg}x={\cos }^{2}x.$$ Then, ${\sin }^{2}x={\cos }^{3}x$ and if we denote by $t=\cos x\in (0;1)$, then we obtain the equation: $$1-t^2=t^3\Rightarrow f(t)=t^3+t^2-1=0.$$