= Solution
Factor the leading term out of the <Maclaurin series> for the <sine>:
$$
\sin x=x\left(1-\frac{x^2}{6}+\frac{x^4}{120}+O(x^6)\right).
$$
The <binomial theorem> gives, through degree four inside the bracket,
$$
\left(1-\frac{x^2}{6}+\frac{x^4}{120}+O(x^6)\right)^6=1+6\left(-\frac{x^2}{6}+\frac{x^4}{120}\right)+\binom62\frac{x^4}{36}+O(x^6).
$$
The coefficient of $x^4$ is $1/20+5/12=7/15$. Multiplying back by $x^6$ yields
$$
\boxed{\sin^6x=x^6-x^8+\frac7{15}x^{10}+O(x^{12}).}
$$
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