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Question Number 187559 by Mingma last updated on 18/Feb/23

Answered by cortano12 last updated on 19/Feb/23

$$\frac{\sin x}{\sin y}.\frac{\cos y}{\cos x}=\frac{1}{3}\Rightarrow \frac{\tan x}{\tan y}=\frac{1}{3}$$$$3\tan x=\tan y$$$$\Rightarrow \tan (x+y)=\frac{4\tan x}{1-3\tan {}^{2}x}=\pm \sqrt{15}$$$$\Rightarrow 2\sin x=\sin y;2\cos x=3\cos y$$$$\Rightarrow 4=\sin {}^{2}y+9\cos {}^{2}y$$$$\Rightarrow 4=\sin {}^{2}y+9-9\sin {}^{2}y$$$$\Rightarrow \sin {}^{2}y=\frac{5}{8}\wedge \cos {}^{2}y=\frac{3}{8}$$$$\Rightarrow \tan y=\pm \sqrt{\frac{5}{3}}$$

Commented by Mingma last updated on 19/Feb/23

Great man!

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