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

Problem

sin(2*x)=sin(x)

Solution

1. Apply the double angle identity for the sine function on the left side of the equation.

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

2. Substitute this identity back into the original equation to set the expression equal to the right side.

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

3. Subtract sin(x) from both sides to set the equation to zero.

2*sin(x)*cos(x)−sin(x)=0

4. Factor out the common term sin(x) from the expression.

sin(x)*(2*cos(x)−1)=0

5. Solve for x by setting each factor equal to zero.

sin(x)=0

cos(x)=1/2

6. Determine the solutions for x within a standard interval like [0,2*π)

x=0,π,π/3,(5*π)/3

7. Conclude that since the equation is only true for specific values of x it is a conditional equation rather than a trigonometric identity.

Final Answer

sin(2*x)=sin(x)* is a conditional equation, not an identity.

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