Verify the Identity sin(2x)+sin(x)=0

Problem

sin(2*x)+sin(x)=0

Solution

1. Apply the double angle identity for sine, which states sin(2*x)=2*sin(x)*cos(x) to rewrite the first term.

sin(2*x)+sin(x)=0

2*sin(x)*cos(x)+sin(x)=0

2. Factor out the common term sin(x) from the left side of the equation.

sin(x)*(2*cos(x)+1)=0

3. Apply the zero product property by setting each factor equal to zero to find the values of x that satisfy the identity.

sin(x)=0

2*cos(x)+1=0

4. Solve for x in the first equation. The sine function is zero at integer multiples of π

x=n*π

5. Solve for x in the second equation by isolating the cosine term.

2*cos(x)=−1

cos(x)=−1/2

6. Determine the angles where the cosine is −1/2 which occur in the second and third quadrants.

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

sin(2*x)+sin(x)=sin(x)*(2*cos(x)+1)=0

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