Can you prove recurrent relations on integrals by taking derivative of both sides of the equation?

Question

As an example let$$\qquad I_n = \int x^n\cos x\,dx\qquad(1).$$ By performing integration by parts we can prove the following: $$I_n = x^n\sin x+ nx^{n–1}\cos x– n(n – 1)I_{n–2}\qquad (2).$$ However, my question is if we are told to prove this relation can we take the derivative of both sides of $(2)$ to prove it? And if so would I say that this proof is by the fundamental theorem of calculus?