已知函数$g(x)=\cos x+\sin x$,$h(x)=\sin\left(x+\dfrac{\pi}{2}\right)+\sin(x+\pi)$,设
$f(x)=g\left(x-\dfrac{\pi}{6}\right)h\left(x-\dfrac{\pi}{6}\right)$,则$f(x)$的单调递增区间是（ ）.
A. $\left[k\pi-\dfrac{\pi}{3},k\pi+\dfrac{\pi}{6}\right](k\in\mathbf{Z})$
B. $\left[k\pi+\dfrac{5\pi}{12},k\pi+\dfrac{11\pi}{12}\right](k\in\mathbf{Z})$
C. $\left[k\pi-\dfrac{\pi}{12},k\pi+\dfrac{5\pi}{12}\right](k\in\mathbf{Z})$
D. $\left[k\pi+\dfrac{\pi}{6},k\pi+\dfrac{2\pi}{3}\right](k\in\mathbf{Z})$