The number of roots of the equation $e^x\sin x=1$ between any two roots of the equation $e^x\cos x=1$ is  

 Let $\mathrm{f}\left(\mathrm{x}\right)=\cos \mathrm{x}\cdot {\mathrm{e}}^{\mathrm{x}}\text{ and }\mathrm{f}\left(\mathrm{\alpha }\right)=\mathrm{f}\left(\mathrm{\beta }\right)=0$
By using  Rolle’s theorem   ${\mathrm{f}}^{\mathrm{\prime }}\left(\mathrm{x}\right)=0\Rightarrow (\sin \mathrm{x}){\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{\mathrm{x}}\cos \mathrm{x}$
$\mathrm{\alpha }<\mathrm{c}<{\mathrm{\beta e}}^{\mathrm{x}}\cos \mathrm{x}={\mathrm{e}}^{\mathrm{x}}\sin \mathrm{x}1={\mathrm{e}}^{\mathrm{c}}\sin \mathrm{c}$