This only takes two lines:
[MATH]\sin x^3=x^3-\frac{x^9}{6}+O(x^{15})\hspace2ex[/MATH](substituting [MATH]x^3[/MATH] for [MATH]x[/MATH] in the expansion of [MATH]\sin x[/MATH])
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You made a second mistake in your first attempt, besides using the wrong series; and you made the same error again here! \(\left(x^3\right)^2 \ne x^{(3^2)}\), and \(\left(x^3\right)^3 \ne x^{(3^3)}\)!
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