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Found problems: 2

2020-2021 OMMC, 9
Tags: trig , trigonometry
The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$ over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$

2000 Moldova National Olympiad, Problem 1
Tags: trigonometry , algebra , trig
Suppose that real numbers $x,y,z$ satisfy $$\frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=p.$$Prove that $\cos(x+y)+\cos(y+z)+\cos(x+z)=p$.