$\text{ If }\frac{3-{\tan }^2\frac{\mathrm{\pi }}{7}}{1-{\tan }^2\frac{\mathrm{\pi }}{7}}=\mathrm{\lambda cos}\frac{\mathrm{\pi }}{7},\text{ then }\mathrm{\lambda }\text{ is }$________.

$\begin{array}{l}\text{ Let }\mathrm{\theta }=\frac{\mathrm{\pi }}{7}\Rightarrow 3\mathrm{\theta }=\mathrm{\pi }-4\mathrm{\theta }\\ \Rightarrow \sin 3\mathrm{\theta }=\sin 4\mathrm{\theta }\\ \Rightarrow 3\sin \mathrm{\theta }-4{\sin }^3\mathrm{\theta }=4\sin \mathrm{\theta cos}\mathrm{\theta cos}2\mathrm{\theta }\\ \Rightarrow 3-4{\sin }^2\mathrm{\theta }=4\cos \mathrm{\theta }\left(2{\cos }^2\mathrm{\theta }-1\right)\\ \end{array}$

$\begin{array}{l}\Rightarrow 4\cos \mathrm{\theta }\left(2{\cos }^2\mathrm{\theta }-1\right)=4{\cos }^2\mathrm{\theta }-1\\ \Rightarrow 4\cos \mathrm{\theta }=\frac{4{\cos }^2\mathrm{\theta }-1}{2{\cos }^2\mathrm{\theta }-1}=\frac{3-{\tan }^2\mathrm{\theta }}{1-{\tan }^2\mathrm{\theta }}\\ \Rightarrow 4\cos \frac{\mathrm{\pi }}{7}=\frac{3-{\tan }^2\frac{\mathrm{\pi }}{7}}{1-{\tan }^2\frac{\mathrm{\pi }}{7}}\end{array}$