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 $$\left\{\alpha (\theta): \theta \in [0, 2\pi)\right\}$$ and
$$\beta^*$$ is the minimum of the set $$\left\{\beta (\theta): \theta \in [0, 2\pi)\right\}$$,
then the value of $$\alpha^* + \beta^*$$ is