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

Problem

sin(2*x)=cos(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. Set the equation to zero by subtracting cos(x) from both sides to find the values of x where the identity holds.

2*sin(x)*cos(x)−cos(x)=0

3. Factor out the common term cos(x) from the expression.

cos(x)*(2*sin(x)−1)=0

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

cos(x)=0⇒x=π/2+n*π

2*sin(x)−1=0⇒sin(x)=1/2⇒x=π/6+2*n*π,(5*π)/6+2*n*π

5. Conclude that the statement sin(2*x)=cos(x) is a conditional equation, not a trigonometric identity, because it is only true for specific values of x

Final Answer

sin(2*x)=cos(x)* is not an identity

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