If $1+\sin x+\cos x\neq 0$ the value of $x$ for which $\begin{vmatrix}1&\sin x&\cos x\\\sin x&1&\cos x\\\cos x&\sin x&1\end{vmatrix}=0$ is

Question

$\begin{array}{1 1}(A)\;n\pi+(-1)^n\large\frac{\pi}{2},\normalsize 2n\pi&(B)\;n\pi,2n\pi\pm \large\frac{\pi}{2}&\\(C)\;n\pi+(-1)^n\large\frac{\pi}{2},\normalsize 2n\pi\pm \large\frac{\pi}{2}&(D)\;\text{None of these}\end{array}$