The equation $\left(\mathrm{cosP}-1\right){\mathrm{x}}^2+\left(\mathrm{cosP}\right)\mathrm{x}+\mathrm{sinP}=0$ where x is a variable has real roots. Then the interval of P may be any of the following :

$$\begin{gathered} (\cos \mathrm{P}-1){\mathrm{x}}^2+\cos \mathrm{P} \mathrm{x}+\sin \mathrm{P}=0 \\ \mathrm{D}\ge 0 \Rightarrow {\mathrm{b}}^2-4\mathrm{ac}\ge 0 \\ {(\cos \mathrm{P})}^2-4\sin \mathrm{P}(\cos \mathrm{P}-1)\ge 0 \\ {\cos }^2\mathrm{P}-4\sin \mathrm{P} \cos \mathrm{P}+4\sin \mathrm{P}\ge 0 \\ ∆<0 \Rightarrow {(4\sin \mathrm{P})}^2-4(4\sin \mathrm{P})<0 \\ \sin \mathrm{P}(\sin \mathrm{P}-1)<0 \\ \sin \mathrm{P}\in (0, 1) \\ \mathrm{P}\in (0, \mathrm{\pi }) \end{gathered}$$