已知函数 $f(x)=\cos 2 x, g(x)=\sin x$, 则存在 $\theta \in\left(\frac{\pi}{6}, \frac{\pi}{4}\right)$, 使得
$\text{A.}$ $2 g(\theta)=f(\theta)+g(\theta) \cdot f(\theta)$
$\text{B.}$ $4 g(\theta) \cdot f(\theta)=f(\theta)+2 g(\theta)$
$\text{C.}$ $2 f(\theta)=g(\theta)+g(\theta) \cdot f(\theta)$
$\text{D.}$ $f(\theta)=g(\theta)$