Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Write down each of the terms in the expansion of $\sin(x^2)-(\sin(x))^2$. Taylor's Theorem applies at the point $a=0$ and with $n=4$.

Got no idea how to proceed. My lecture notes have one example that I barely understand. I'd really appreciate a semi-detailed overview of what I should be doing. I can go to class and ask questions, but I need to have some idea of what I am looking at in order to ask intelligent questions.

we will keep track of upto powers of $x^4.$  $$\sin(x) = x - \frac 16 x^3 + \cdots, (\sin x)^2 = x^2 - \frac 13 x^4 + \cdots, \sin(x^2) = x^2+ 0x^4 + \cdots $$ therefore $$\sin (x^2) - (\sin x)^2 = \frac 13 x^4 + \cdots $$
