Explanation:

- To find the derivative of \(\sin^2 x\) with respect to \(\cos^2 x\), we can apply implicit differentiation.

- Let \(u = \sin^2 x\) and \(v = \cos^2 x\).

- Note that \(\sin^2 x + \cos^2 x = 1\).

- Differentiating both sides with respect to \(x\), we get:

- \(\frac{d}{dx}(\sin^2 x) + \frac{d}{dx}(\cos^2 x) = 0\).

- The derivative of \(\sin^2 x\) is \(2\sin x \cos x\).

- The derivative of \(\cos^2 x\) is \(-2\sin x \cos x\).

- Therefore, using chain rule:

- \(\frac{du}{dv} = \frac{2\sin x \cos x}{-2\sin x \cos x} = -1\).

- Option 1: -1 is the correct answer.

- Option 2: 1, is not correct.

- Option 3: \(\sin 2x\), refers to \(2\sin x \cos x\) which is not the derivative.

- Option 4: \(\cos 2x\), is unrelated to the calculation.

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