Inverse Trigonometric Fun question bank (Maths)

Considering the principal values of the inverse trigonometric functions, $\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^2}\right),-\frac{1}{2} \lt x \lt \frac{1}{\sqrt{2}}$, is equal to

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta-8$ and $\sec ^2\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^2\left(\cot ^{-1} \beta\right)-36$, then $\alpha^2+\beta$ is_______.