If the equation ${\cos }^4\mathrm{\theta }+{\sin }^4\mathrm{\theta }+\mathrm{\lambda }=0$ has a real solution for $\mathrm{\theta }$, then $\mathrm{\lambda }$ lies in the interval

$\begin{array}{l}{\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }+\mathrm{\lambda }=0\Rightarrow -\mathrm{\lambda }={\sin }^4\mathrm{\theta }+{\cos }^4\mathrm{\theta }={\left({\sin }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }\right)}^2-2{\sin }^2{\mathrm{\theta cos}}^2\mathrm{\theta }=1-\frac{{\sin }^{2}2\mathrm{\theta }}{2}\\ \Rightarrow \mathrm{\lambda }=-1+\frac{1}{2}{\sin }^{2}2\mathrm{\theta }\in \left[-1,-\frac{1}{2}\right]\end{array}$