I was attempting a question asked here
Evaluate: $$\lim_{x\to0}\frac{x\sin{(\sin{x})}-\sin^2{x}}{x^6}$$
And the answers involved the expansion of $\sin(\sin(x))$. While the coefficients for $x$ and $x^3$ match with my attempt, unfortunately, the coefficient of $x^5$ doesn't:
$$\sin(\sin (x))=x-\frac{x^3}{3}+\frac {x^5}{10} + O(x^6)$$
My attempt:
$$S(x)= \sin(\sin(x)) =\left (\sin x -\frac{\sin^3 x}{6} +\cdots\right )$$
Now using the expansion for $\sin x$, $$S(x)=\left(x-\frac{x^3}{6}+\frac{x^5}{120}+\cdots\right)-\frac 16\left(x-\frac{x^3}{6}+\frac{x^5}{120}+\cdots\right)^3+\cdots$$
Evaluating this gives me $$S(x)=x-\frac{x^3}{3} +\frac{11x^5}{120} +O(x^6)$$
As I'm new to handling such stuff, could anyone please tell me where I missed a $\frac{x^5}{120}$?
P.S. This post doesn't help me as it doesn't need the fifth power.
