The maximum value of

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

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

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

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

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

Expanding along $C_{1}$

$=1(\sin \theta \cos \theta)$

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

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

But, $\sin 2 \theta \leq 1$

$\Rightarrow \frac{\sin 2 \theta}{2} \leq \frac{1}{2}$

Hence, the maximum value of $\left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & 1+\sin \theta & 1 \\ 1 & 1 & 1+\cos \theta\end{array}\right|$ is $\underline{\frac{1}{2}}$.