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Question Number 35886 by behi83417@gmail.com last updated on 25/May/18

Answered by ajfour last updated on 25/May/18

$$\sin (x+y)-\sin x=\sin y$$$$2\cos (x+\frac{y}{2})\sin \frac{y}{2}=2\cos \frac{y}{2}\sin \frac{y}{2}$$$$\Rightarrow \sin \frac{y}{2}[\cos (x+\frac{y}{2})-\cos \frac{y}{2}]=0$$$$or\left(\sin \frac{y}{2}\right)\left(\sin \frac{x}{2}\right)\sin \left(\frac{x+y}{2}\right)=0$$$$\Rightarrow y=2m\pi ,orx=2n\pi ,or$$$$x+y=2k\pi$$$$andsincex-y=\frac{\pi }{3}wehave$$$$x=2n\pi ,y=2n\pi -\frac{\pi }{3}$$$$ory=2m\pi ,x=2m\pi +\frac{\pi }{3}$$$$or$$$$x=k\pi +\frac{\pi }{6},y=k\pi -\frac{\pi }{6}.$$

Commented by behi83417@gmail.com last updated on 25/May/18

$$thankyouverymuchsirAjfour.$$

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