If $\Delta \left(x\right)=\left|\begin{array}{ccc}1& \cos x& 1-\cos x\\ 1+\sin x& \cos x& 1+\sin x-\cos x\\ \sin x& \sin x& 1\end{array}\right|$ then ${\int }_0^{\pi /2}\Delta \left(x\right)dx$ equal 

Using , $C_1\to C_1-C_2-C_3,$ we get 

$\Delta \left(x\right)=\left|\begin{array}{ccc}0& \cos x& 1-\cos x\\ 0& \cos x& 1+\sin x-\cos x\\ -1& \sin x& 1\end{array}\right|$

$=(-1)\left|\begin{array}{cc}\cos x& 1-\cos x\\ \cos x& 1+\sin x-\cos x\end{array}\right|$

$=(-1)\cos x[1+\sin x-\cos x-1+\cos x]$

$=-\cos x\sin x=-\frac{1}{2}\left(\sin 2x\right)$

${\left.\therefore {\int }_0^{\pi /2}\Delta \left(x\right)dx=-\frac{1}{2}\frac{(-1)}{2}\cos 2x\right]}_0^{\pi /2}$

$=\frac{1}{4}(\cos \pi -\cos 0)=-\frac{1}{2}$