If the equation \(\cos ^4 \theta+\sin ^4 \theta+\lambda=0\). has real solutions for \(\theta\), then \(\lambda\) lies in the interval

\(\begin{aligned}
& \cos ^4 \theta+\sin ^4 \theta+\lambda=0 \\
& \Rightarrow\left(\sin ^2 \theta+\cos ^2 \theta\right)^2-2 \sin ^2 \theta \cos ^2 \theta+\lambda=0 \\
& \Rightarrow 1-2 \sin ^2 \theta \cos ^2 \theta+\lambda=0 \\
& \Rightarrow \lambda=2 \sin ^2 \theta \cos ^2 \theta-1 \\
& \Rightarrow \lambda=\frac{\sin ^2 2 \theta}{2}-1
\end{aligned}\)
Since \(-1 \leq \sin 2 \theta \leq 1\),
\(\begin{aligned}
& 0 \leq \sin ^2 2 \theta \leq 1 \\
& \Rightarrow 0 \leq \frac{\sin ^2 2 \theta}{2} \leq \frac{1}{2} \\
& \Rightarrow-1 \leq \frac{\sin ^2 2 \theta}{2}-1 \leq-\frac{1}{2} \\
& \Rightarrow \lambda \in\left[-1,-\frac{1}{2}\right]
\end{aligned}\)