You cannot simply substitute the maximum value of sin to get the maximum value of the function.

First of all, you should know the triple angle identity: $\displaystyle 3\sin y - 4 \sin^3 y = \sin 3y$,

This is what you had to do:

$\displaystyle 9\sin \frac{10x}{3} - 12 \sin^3 \frac{10x}{3} = 3 \left(3\sin \frac{10x}{3} - 4 \sin^3 \frac{10x}{3}\right)$

Now use the triple angle identity,

$\displaystyle 3 \left(3\sin \frac{10x}{3} - 4 \sin^3 \frac{10x}{3}\right) = 3 \sin 3\left(\frac{10x}{3}\right) = 3\sin 10 x$

Greatest value of $\displaystyle 3\sin 10 x$ is 3 and occurs when $\displaystyle \sin 10 x = 1$

So now you will get the answer

$\displaystyle 3\sin 6x - 4\sin 2x = 0$

Now use the $\displaystyle \sin 3x = 3 \sin x - 4 \sin^3 x$ identity, with x replaced by 2x.

$\displaystyle 3(3 \sin 2x - 4 \sin^3 2x ) - 4 \sin 2x = 0$

So can you continue now?