>>12285021(i) Integrate by parts and trig pythagoras
(ii) Induction
(iii) , double angle formula
(iv) I_n is decreasing from (i), or alternatively the J_n integrand is everywhere less than 1/4.
The only thing about this question that isn't just rote calculation is the substitution for part (iii), but even that is a standard step for the beta function.
But because it's all standard techniques, it's a good weed-out check for who's got the basics down enough to learn proper math later.
It's only difficult for math-savy HS students due to lack of practice. Even freshman's first uni analysis course has (some) more difficult questions than this, because they atleast require some modicum of creative thought.
