sin2x = sin x / solve for x please show me how

Question

sin2x = sin x / solve for x please show me how to solve for x.

Answer 1

sin(2x) = sinx

 

2sinxcosx - sinx = 0

 

sinx(2cosx - 1) = 0

 

sinx = 0 or cosx = 1/2

 

If sinx = 0, then x = kπ, k = 0, ±1, ±2,...

 

If cosx = 1/2, then x = π/3 + 2kπ, 5π/3 + 2kπ, k = 0, ±1, ±2, ...

 

If you just need the solutions in the interval [0, 2π], they are:

 

   0, π, 2π, π/3, 5π/3

 

 

Answer 2

sin 2x = sin x

2 sin x cos x = sin x          double angle formula

2 sin x cos x - sin x = 0     subtract sin x from both sides

sin x (2 cos x - 1) = 0       factor out sin x

By Zero Product Rule

sin x  = 0, i.e., x = kπ where k is any integer

Or

2 cos x - 1 = 0

2 cos x = 1         add 1 to both sides

cos x = 1/2

x = π/6 + 2kπ where k is any integer

or 

x = 5π/6 + 2kπ where k is any integer