There is a full solution available online (for those interested) so I shan't repeat it - as it is also very long.
However, the key insight to solve that seemingly monstrous product is to use one of the trig identities - but in an unusual way - so that within just 2 or 3 terms the mind can see that the infinite product simplifies to just two terms.
Such problems are very much borderline A Level Further Mathematics - scanning the MIT paper, I think at least half of the problems could be done at school. Nowadays, a student would have to be studying Cambrdge's new Pre-U Level Mathematics as they have brought back all the topics that they have slowly removed from the (dumbed down) A Level.
RE: Can you solve this integral from MIT's Integration Bee?