Let$$ I_n =\int_0 ^\frac{\pi}{2} \cos^nx\cos nx\, dx$$ Express $I_n$ in terms of $I_{n-1}$
I had a thought about the possibility of using complex numbers in this. I'm in highschool and this is something obviously not taught and neither do I know the proper theory , and one of my friends told me about this possibility.
What I thought was to write it as $Re\{\int_0 ^\frac{\pi}{2} \cos^nx e^{inx}dx\}$ but after this I could not think of any way that could help. Is it really a dead end ?? Is this even correct?? Could something be done after this ?? Is this a way/method of evaluation of integrals ?? (Just like eg substitution) If yes , is there a place where I could look up the proper theory for it.
Even though the solution itself is as easy as applying by parts and manipulation, but I just want to know if my initial thought could also have lead to the same result.