Find tan⁡(x+y) given sin⁡x=3sin⁡(x+2y)My reference gives the solution −2tan⁡y but how do I prove it ?My Attempt3=sin⁡(x+2y)sin⁡x⇒3sin⁡x=sin⁡x.cos⁡2y+cos⁡x.sin⁡2y3sin⁡x=2sin⁡x.cos2y−sin⁡x+2cos⁡x.sin⁡y.cos⁡y=2cos⁡y.sin⁡(x+y)−sin⁡x⇒4sin⁡x=2cos⁡y.sin⁡(x+y)⇒sin⁡(x+y)=2.sin⁡xcos⁡ycos⁡(x+y)=sin⁡xsin⁡y⇒tan⁡(x+y)=2tan⁡y

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Hint:As we need to eliminate xlet x+y=k,  x=k−ysin⁡(y−k)=3sin⁡(k−y+2y)Method#1:Expand sin⁡(y−k),sin⁡(y+k) and group sin⁡k,cos⁡kMethod#2:sin⁡(y−k)sin⁡(y+k)=31

Find \(\displaystyle{\tan{{\left({x}+{y}\right)}}}\) given \(\displaystyle{\sin{{x}}}={3}{\sin{{\left({x}+{2}{y}\right)}}}\) My reference gives

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