The value of $\frac{{\mathrm{e}}^{-\mathrm{\pi }/4}+{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}{\tan }^{50}\mathrm{xdx}}{{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}\left({\tan }^{49}\mathrm{x}+{\tan }^{51}\mathrm{x}\right)\mathrm{dx}}$ is

$$\begin{gathered} \begin{array}{l}{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}\left({\mathrm{Tan}}^{49}x+{\mathrm{Tan}}^{51}\mathrm{x}\right)\mathrm{dx} \\ ={\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}{\mathrm{Tan}}^{49}\mathrm{x}\left({\sec }^2\mathrm{x}\right)\mathrm{dx} \\ ={\left({\mathrm{e}}^{-\mathrm{x}}\cdot \frac{{\tan }^{50}x}{50}\right)}_0^{\frac{\mathrm{\pi }}{4}}+{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}\frac{{\mathrm{Tan}}^{50}\mathrm{x}}{50}\mathrm{dx}\\ =\frac{{\mathrm{e}}^{-\frac{\mathrm{\pi }}{4}}}{50}+\frac{1}{50}{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}{\mathrm{Tan}}^{50}\mathrm{xdx}\\ =\frac{{\mathrm{e}}^{\frac{-\mathrm{\pi }}{4}}+{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}{\mathrm{Tan}}^{50}\mathrm{xdx}}{50}\\ 50=\frac{{\mathrm{e}}^{\frac{-\mathrm{\pi }}{4}}+{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}{\mathrm{Tan}}^{50}\mathrm{xdx}}{{\int }_0^{\frac{\mathrm{\pi }}{4}} {\mathrm{e}}^{-\mathrm{x}}\left({\mathrm{Tan}}^{49}x+{\mathrm{Tan}}^{51}\mathrm{x}\right)\mathrm{dx}}\end{array} \end{gathered}$$