Official solution
cos4 \[\theta \] + sin4 \[\theta \] +
\[\lambda \] = 0
\[ \Rightarrow \] 1 -- 2sin2 \[\theta \] cos2 \[\theta \] = -\[\lambda \]
\[ \Rightarrow \] 1 - \[\frac{1}{2} \times 4\]sin2 \[\theta \] cos2 \[\theta \] = -\[\lambda \]
\[ \Rightarrow \] 1 - \[\frac{\sin^{2} 2\theta }{2} \] = -\[\lambda \]
\[ \Rightarrow \]2(\[\lambda \] + 1) = sin2 2\[\theta \]
0 \[ \le \] 2 (\[\lambda \] + 1) \[ \le \] 1
0 \[ \le \] (\[\lambda \] + 1) \[ \le \] \[\frac{1}{2} \]
-1 \[ \le \] \[\lambda \] \[ \le \] -\[\frac{1}{2} \]