question_answer

If \[\Delta (x)=\left| \begin{matrix}    1 & \cos x & 1-\cos x  \\    1+\sin x & \cos x & 1+\sin x-\cos x  \\    \sin x & \sin x & 1  \\ \end{matrix} \right|,\] then \[\int_{0}^{\pi /2}{\Delta (x)\,\,dx}\] equals

A)  \[\frac{1}{4}\]                         

B)  \[\frac{1}{2}\]

C)  \[0\]                            

D) \[-\frac{1}{2}\]

E)  None of these