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(AM = Arithmetic mean).The options given are $x>y$ , $x =y$ , $2x=y$ , $x=2y$. The correct answer is $2x = y$. How do you solve this? I tried a bit of rough work and got $y=cosec40$. For $x$, it tried writing the equation in terms of angle 20 degree, so I wrote $\tan50 =\tan (30+20)$ , expanded it and got an equation in terms of tan and x only. Don't know what to do further. Don't even know if my current approach is correct. Please help further with my approach a suggest a Better one!

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Logically $y>x$, because $\tan{70}>\tan{50}$, hence the choice $y=2x$ is right.

Proof: $$x=\frac{\tan{20}+\tan{50}}{2}=\frac{\sin{(20+50)}}{2\cos{20} \cos{50}}=\frac{1}{2\cos{50}}.$$ $$y=\frac{\tan{20}+\tan{70}}{2}=\frac{\sin{(20+70)}}{2\cos{20}\cos{70}}=\frac{1}{\sin{40}}=\frac{1}{\cos{50}}.$$

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