If ${\tan }^2\mathrm{\theta }=2{\tan }^2\mathrm{\varphi }+1$ then ${\sin }^2\mathrm{\varphi }+\cos 2\mathrm{\theta }=$

$\begin{array}{l}{\tan }^2\mathrm{\theta }=2{\tan }^2\mathrm{\varphi }+1\\ \text{ or }1+{\tan }^2\mathrm{\theta }=2\left(1+{\tan }^2\mathrm{\varphi }\right)\\ \Rightarrow {\sec }^2\mathrm{\theta }=2{\sec }^2\mathrm{\varphi }\\ \Rightarrow {\cos }^2\mathrm{\varphi }=2{\cos }^2\mathrm{\theta }=1+\cos 2\mathrm{\theta }\\ \Rightarrow \cos 2\mathrm{\theta }={\cos }^2\mathrm{\varphi }-1=-{\sin }^2\mathrm{\varphi }\\ \Rightarrow {\sin }^2\mathrm{\varphi }+\cos 2\mathrm{\theta }=0\end{array}$