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Consider the following statements: Statement 1: \(\sin^{-1}\!\left(\frac{1}{\sqrt{e}}\right) > \tan^{-1}\!\left(\frac{1}{e}\right)\) Statement 2: For any \(x, y \in (0,1)\), if \(x > y\), then \(\sin^{-1}x > \tan^{-1}y\).

1. Statement 1 is true, Statement 2 is true, and Statement 2 correctly explains Statement 1.
2. Statement 1 is true, Statement 2 is true, but Statement 2 does not correctly explain Statement 1.
3. Statement 1 is false, Statement 2 is true.
4. Statement 1 is true, Statement 2 is false.

Correct answer: Statement 1 is false, Statement 2 is true.

Solution

Statement 1 is false because the value of \\sin^{-1}\\left(\\frac{1}{\\sqrt{e}}\\right) is less than \\tan^{-1}\\left(\\frac{1}{e}\\right}, while Statement 2 is true as it holds that for any x and y in (0,1), if x is greater than y, then \\sin^{-1}x is indeed greater than \\tan^{-1}y.

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