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Let $M=\left[\begin{array}{cc}\sin ^4 \theta & -1-\sin ^2 \theta \\ 1+\cos ^2 \theta & \cos ^4 \theta\end{array}\right]=\alpha I +\beta M ^{-1}$, where $\alpha=\alpha(\theta)$ and $\beta=\beta(\theta)$ are real number, 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
   $-\frac{37}{16}$
• ✓
  $-\frac{29}{16}$
• C
  $-\frac{31}{16}$
• D
  $-\frac{17}{16}$