I need to solve this limit without using L'Hospital's rule

Question

I need to solve this limit without using L'Hospital's rule.

$$\lim_{x\to 0}⁡(\cos x)^\left(\frac{-4}{x^2}\right)⁡$$

Answer 1

Hint:

$$(\cos x)^{-4/x^2}=(1-\sin^2x)^{-2/x^2}=\left(\left(1-{1\over\csc^2x} \right)^{\csc^2x}\right)^{-2\left(\sin x\over x\right)^2}$$

Can you (and/or your friend) take it from there?