Solve $\sin 7x+\sin 3x=0$

Question

I tried it by graph plotting, but it is going so ugly. On solving it on paper it mixed up. Is there any other process to solve?

Answer 1

Hint: by addition formulas you have $$\sin (5x+2x) = \sin 5x \cos 2x + \cos 5x \sin 2x$$ $$\sin (5x-2x) = \sin 5x \cos 2x - \cos 5x \sin 2x$$ so if you add the two equations you end up with $$\sin (5x+2x) + \sin (5x-2x) = 2\sin 5x \cos 2x.$$

Answer 2

Hint:

Use Prosthaphaeresis Formulas

and $\sin y=0\implies y=m\pi$

and $\cos z=0\implies z=(2r+1)\dfrac\pi2$

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OR

$$\sin7x-\sin3x=\sin(-3x)\implies7x=n\pi+(-1)^n(-3x)$$ where $m,n,r$ are arbitrary integers

Can you show equivalence of the two results?