There is a question on this website which tries to prove that the value of nth derivative of $\cos(x)$ is 'something'.
The question is given below:
Showing the $n$-th derivative of $\cos x$ by induction
What if I were to find the value of $n$-th derivative rather than prove it using induction and other methods.
How to 'FIND THE VALUE OF IT'?
Try the first $n^{th}$ derivatives and observe the pattern. You get
$$\cos x,-\sin x,-\cos x,\sin x$$ and so on periodically.
If you observe this on the trigonometric circle, you will notice rotations by $\frac\pi2$ radians each time, so that you can summarize as
$$\cos\left(x+n\frac\pi2\right).$$
A more "advanced" way is by means of complex numbers. We have
$$\cos x=\Re e^{ix},$$
then
$$(e^{ix})^{(n)}=i^ne^{ix}=e^{i(x+n\pi/2)}.$$