Answer
$-cosx$
Work Step by Step
To find the pattern, first calcluate the first 4 derivatives of $sinx$ using the trig derivative identities. If $y=sinx$, then it follows that $y'=cosx$,$y''=-sinx$, and $y^{(4)}=sinx$. Here, it is obvious that the derivatives will repeat, so now we just need to know which one of the 4 the 99th derivative is. When we divide $\frac{99}{4}$, we get a remainer of 3, which indicates that $y^{(99)}=y'''$. It then follows that $y^{(99)}=-cosx$.
