$\large \lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2x} = \frac{A}{B}$

The limit above holds true for coprime integers $A$ and $B$, where $A$ is negative and $B$ is positive. Find $A+B$.

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**Bonus**: Evaluate this limit without applying L'Hôpital's rule.