$3 \sin x + 4 \cos y = 5$, $4 \sin y + 3 \cos x = 2$ How to find $\sin x$, $\sin y$, $\cos x$, $\cos y$, 2020 contest question https://www.cemc.uwaterloo.ca/contests/past_contests/2020/2020CSMC.pdf
https://www.cemc.uwaterloo.ca/contests/past_contests/2020/2020CSMCSolution.pdf
Question 5 from 2020 CSMC math contest:
$$3 \sin x + 4 \cos y = 5,$$ $$4 \sin y + 3 \cos x = 2.$$
Find $\sin(x+y)$.
The solution page added that it is also possible to find $\sin x$, $\sin y$, $\cos x$, $\cos y$ with a different approach but I could not figure it out.

We have $4\cos y=5-3\sin x$.  Similarly we have $4\sin y=2-3\cos x$.  Squaring and add:
$$
16=(5-3\sin x)^2+(2-3\cos x)^2=38-30\sin x-12\cos x
$$
so $15\sin x+6\cos x=11$
which gives you possible $\sin x,\cos x$ (i.e., solve the system $15\sin x+6\cos x=11, \sin^2 x+\cos^2 x=1$ to get $\sin x=\frac{55\pm 4\sqrt{35}}{87}$, and similar for $\cos x$, or solve for $\tan\frac12x=\frac{15\pm 2\sqrt{35}}{17}$) and hence $\sin y,\cos y$.
