Let $M=\begin{bmatrix} \sin^4 \theta & -1-\sin^2 \theta \\ 1+\cos^2 \theta & \cos^4 \theta \end{bmatrix} = \alpha I + \beta M^{-1}$,where $\alpha = \alpha(\theta)$ and $\beta = \beta(\theta)$ are real numbers,and $I$ is the $2 \times 2$ identity matrix. If $\alpha^*$ is the minimum of the set $\{\alpha(\theta) : \theta \in [0, 2\pi)\}$ and $\beta^*$ is the minimum of the set $\{\beta(\theta) : \theta \in [0, 2\pi)\}$,then the value of $\alpha^* + \beta^*$ is

• A
  $-\frac{37}{16}$
• B
  $-\frac{29}{16}$
• C
  $-\frac{31}{16}$
• D
  $-\frac{17}{16}$