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The given limit is:

\[
\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x}
\]

Let's solve this step by step.

### Step 1: Expand \(\cos x\) using the Taylor series around \(x = 0\).

For small values of \(x\), we can use the Taylor expansion of \(\cos x\) around 0:

\[
\cos x = 1 - \frac{x^2}{2} + O(x^4)
\]

### Step 2: Expand the square root and cube root of \(\cos x\).

- For \(\sqrt{\cos x}\), we can use the binomial approximation for small \(x\):

\[
\sqrt{\cos x} = \sqrt{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{4} + O(x^4)
\]

- For \(\sqrt[3]{\cos x}\), the binomial expansion gives:

\[
\sqrt[3]{\cos x} = \sqrt[3]{1 - \frac{x^2}{2}} \approx 1 - \frac{x^2}{6} + O(x^4)
\]

### Step 3: Simplify the numerator.

Now, subtract the expansions of \(\sqrt{\cos x}\) and \(\sqrt[3]{\cos x}\):

\[
\sqrt{\cos x} - \sqrt[3]{\cos x} \approx \left(1 - \frac{x^2}{4}\right) - \left(1 - \frac{x^2}{6}\right) = - \frac{x^2}{4} + \frac{x^2}{6}
\]

Simplifying this:

\[
\sqrt{\cos x} - \sqrt[3]{\cos x} \approx - \frac{x^2}{12}
\]

### Step 4: Simplify the denominator.

We know that \(\sin^2 x \approx x^2\) for small \(x\), so:

\[
\sin^2 x \approx x^2
\]

### Step 5: Compute the limit.

Now substitute the simplified expressions back into the limit:

\[
\lim_{x \to 0} \frac{-\frac{x^2}{12}}{x^2} = \lim_{x \to 0} -\frac{1}{12} = -\frac{1}{12}
\]

### Final Answer:

\[
\lim_{x \to 0} \frac{\sqrt{\cos x} - \sqrt[3]{\cos x}}{\sin^2 x} = -\frac{1}{12}
\]

Would you like further clarification on any step?

Here are some related questions to expand your understanding:

1. How do we derive the Taylor series for \(\cos x\)?
2. Can you explain the binomial expansion used for the square and cube roots?
3. What is the approximation for \(\sin x\) as \(x \to 0\)?
4. How does the small-angle approximation simplify trigonometric functions?
5. Could you find the limit using L'Hôpital's Rule instead of series expansion?

**Tip:** When dealing with limits involving trigonometric functions near 0, using Taylor expansions is often a quick way to simplify complex expressions.

Find the limit as x approaches 0 for the expression (sqrt(cos x) - cbrt(cos x)) / sin^2 x.

Explore the detailed calculation of the limit (sqrt(cos x) - cbrt(cos x)) / sin^2 x as x approaches 0 using Taylor series expansion and binomial approximations. This advanced calculus problem delves into the behavior of trigonometric functions near zero.

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