Given
tan(2x)·sin(x) = tan(2x)
Find
x on the interval [0, 2π)
Solution
Subtract the right side and factor. Invoke the zero-product rule.
... tan(2x)sin(x) -tan(2x) = 0
... tan(2x)(sin(x) -1) = 0
... tan(2x) = 0
... 2x = arctan(0) = nπ
... x = (n/2)π . . . . n = {0, 1, 2, 3}
And when the other factor is zero, we have
... sin(x) -1 = 0
... sin(x) = 1
... x = arcsin(1) = π/2
So, we have
... x ∈ {0, π/2, π, 3π/2}