Edward C. answered • 05/14/18

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Caltech Grad for math tutoring: Algebra through Calculus

I'll use a simpler example to explain what's happening. Suppose you were trying to solve the equation x

^{2}= 3x. If you divided both sides by x, you would get the solution x = 3. But you would lose the solution x = 0 (which is obviously a valid solution for the original equation). Why? Because division by 0 is undefined, so you can only divide both sides of an equation by a non-zero value. So anytime you divide both sides of an equation by a variable you must account for the case that the variable might be equal to 0. In fact, it's usually a good idea to try to avoid dividing both sides of an equation by a variable so you don't have to worry about any special cases. You can solve without dividing by bringing all the terms to the left side and factoring: x^{2}- 3x = 0, so x(x - 3) = 0, so x = 0 and x = 3 are the solutions.If you do the same for your problem you will get sin(2x)cos(2x) - sin(3x)sin(2x) = 0, which factors to

[sin(2x)]*[cos(2x) - sin(3x)] = 0

which implies that sin(2x) = 0 or cos(2x) = sin(3x).