Find the maximum value

Let $\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cos \theta\end{array}\right|$

Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1}$, we get

$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 0 & \sin \theta & 0 \\ 0 & 0 & \cos \theta\end{array}\right|$

$=\sin \theta \cos \theta$

$=\frac{\sin 2 \theta}{2}$

We know that $-1 \leq \sin 2 \theta \leq 1$.

$\therefore$ Maximum value of $\Delta=\frac{1}{2} \times 1=\frac{1}{2}$