14 didn't seem too hard. Fairly obvious pigeonhole principle for a). b) was a bit of a difficult mechanics but it was fairly straight forward. c) was changing an expanded compound angle back to the single cos. d) was a bit poorly worded about what it actually wanted you to find. I put down a piecewise for -1 and 1 where it was defined but idk. e) basically impossible unless you realised you needed to expand tan(alpha + beta + gamma) and use sums and products of roots
for the proj motion, i did it once (it was right but it looked wrong), crossed it out and did it again (which didnt work), crossed it out AGAIN and redid it, and then realised my original sol was correct so i wasted like 7 mins on that bruh
and the last one, i got the expansion = -1, but then totally forgot about arctanning it to getr 3pi/4 or whatever so yeah im cooked. scaling gotta carry tbh