L i m i t s !

$Let \sin^{-1}{x} = t$

$=$ $\displaystyle \lim_{t \to 0} \frac{\sin^{-1}{t}\cdot \sin{t} - t^2}{(\sin^{-1}{t})^6}$

$=$ $\displaystyle \lim_{t \to 0} \frac{\sin^{-1}{t}\cdot \sin{t} - t^2}{t^6}\cdot \frac{t^6}{(\sin^{-1}{t})^6}$

$= \displaystyle \lim_{t \to 0}\frac{(t + \frac{t^3}{6} + \frac{9t^5}{120} + \frac{5t^7}{112} + \dots) \cdot (t - \frac{t^3}{3!} + \frac{t^5}{5!}) - t^2}{t^6}$ $\times 1$

$=\displaystyle \lim_{t \to 0} \frac{t^6(\frac{1}{5!}-\frac{1}{6\cdot (3!)} + \frac{9}{120} \dots)}{t^6}$

$=$ $\dfrac{1}{120} +$ $\dfrac{9}{120} -$ $\dfrac{1}{36}$

$=$ $\dfrac{1}{18}$

We can solve for the limit using Maclaurin series as follows.

$\begin{aligned} L & = \lim_{x \to 0} \frac {x\sin(\sin x) -\sin^2 x}{x^6} \\ & = \lim_{x \to 0} \frac {x\left(\sin x - \frac {\sin^3 x}{3!} + \frac {\sin^5 x}{5!}-\cdots\right) - \left(x - \frac {x^3}{3!} + \frac {x^5}{5!} - \cdots \right)^2}{x^6} \\ & = \small \lim_{x \to 0} \frac {x\left(x - \frac {x^3}{3!} + \frac {x^5}{5!} - \cdots - \frac 1{3!}\left(x^3 - \frac {3x^5}{3!} +\cdots \right) + \frac 1{5!}\left(x^5 - \cdots\right)\right) - \left(x^2 - \frac {2x^4}{3!} + \left(\frac 2{5!} + \frac 1{(3!)^2} \right)x^6 - \cdots \right)}{x^6} \\ & = \lim_{x \to 0} \frac {\left(x^2 - \frac 2{3!}x^4 +\left(\frac 2{5!}+\frac 3{(3!)^2}\right)x^6 \right)- \left(x^2-\frac 2{3}x^4 +\left(\frac 2{5!}+\frac 1{(3!)^2}-cdots \right)x^6 - \cdots \right)}{x^6} \\ & = \lim_{x \to 0} \frac {\frac 2{(3!)}x^6-O(x^8)}{x^6} = \lim_{x \to 0} \frac 1{18} - O(x^2) = \boxed{\frac 1{18}} \end{aligned}$