Find the value of $$\dfrac{\cos30.5 + \cos31.5+\cdots + \cos44.5}{\sin30.5 + \sin31.5+\cdots + \sin44.5}$$
This came from HMMT 2018 Problem no 17. Could someone explain to me the solution with a diagram. I solved it in a different way.
https://hmmt-archive.s3.amazonaws.com/tournaments/2018/feb/guts/solutions.pdf
Each sum is telescopic! Show that $$\cos(0.5+k)=\frac{\sin(k+1)-\sin(k)}{2\sin(0.5)} \quad\text{and}\quad \sin(0.5+k)=\frac{\cos(k)-\cos(k+1)}{2\sin(0.5)}.$$ Therefore at the numerator we have $$\cos(30.5) + \cos(31.5)+\cdots + \cos(44.5)=\frac{\sin(45)-\sin(30)}{2\sin(0.5)}.$$ In a similar way at the denominator we find $$\sin(30.5) + \sin(31.5)+\cdots + \sin(44.5) =\frac{\cos(30)-\cos(45)}{2\sin(0.5)}.$$