Matrices and Determinants: JEE Advanced 2019 Mathematics question

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 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