Why can't I cancel sin(3θ) from both sides

Question

Solve: sin(3θ)cos(θ)=sin(3θ)/2, for 0<θ<2π

Answer 1

You can divide by sin(3θ) on both sides but by doing so, you will be eliminating some of the solution. What you are eliminating is the case when sin(3θ) = 0.

To help understand, let's make up a simple problem. (x-1)x = 1/2(x-1). If you divide by (x-1) on both sides to solve, the answer is x = 1/2. But let's multiply it out and solve a different way.

(x-1)x = 1/2(x-1)

x² - x = 1/2x - 1/2

2(x² - x) = 2(1/2x - 1/2)

2x² - 2x = x - 1

2x² - 3x + 1 = 0

(2x - 1)(x - 1) = 0

(2x - 1) = 0 or (x - 1) = 0

x = 1/2 or x = 1

Plugging those numbers in, we see that both are truly correct answers.

So, by dividing by (x-1), the net result was we got the one answer (x = 1/2) but lost the other answer x = 1. Notice that the answer that was lost is when the divisor (x-1) = 0

Answer 2

You cancel from both sides through division. However by canceling out sin(3theta) from both sides you will miss out on possible solutions, whatever makes sin(3theta) equal to 0. Remember, you never want to divide by zero. To solve this, bring everything to one side, factor, and set each factor equal to 0 and solve.  

Try it it yourself and see. 

First, bring everything to one side, factor, and solve.

Second, cancel first then solve. If you do it this way you will see you are missing some solutions you got in the first step.