HMMT 2018 U14

Given that $x$ is a positive real, find the maximum possible value of

$\sin \left(\tan ^{-1}\left(\frac{x}{9}\right)-\tan ^{-1}\left(\frac{x}{16}\right)\right)$

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{7}{25}

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