Solve for x sin(4x)-sin(2x)=0

Problem

sin(4*x)−sin(2*x)=0

Solution

1. Apply the double-angle identity for sin(4*x) by treating it as sin(2⋅2*x)

sin(4*x)=2*sin(2*x)*cos(2*x)

2. Substitute this identity back into the original equation.

2*sin(2*x)*cos(2*x)−sin(2*x)=0

3. Factor out the common term sin(2*x)

sin(2*x)*(2*cos(2*x)−1)=0

4. Set each factor to zero to find the possible solutions for 2*x

sin(2*x)=0

2*cos(2*x)−1=0⇒cos(2*x)=1/2

5. Solve for 2*x using the general solutions for sine and cosine, where n is an integer.

2*x=n*π

2*x=π/3+2*n*π

2*x=−π/3+2*n*π

6. Divide by 2 to isolate x

x=(n*π)/2

x=π/6+n*π

x=−π/6+n*π

Final Answer

sin(4*x)−sin(2*x)=0⇒x=(n*π)/2,π/6+n*π,−π/6+n*π

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