Verify the Identity cos(2x)-cos(x)=0

Problem

cos(2*x)−cos(x)=0

Solution

1. Apply the double angle identity for cos(2*x) to rewrite the equation in terms of cos(x)

cos(2*x)=2*cos^2(x)−1

2. Substitute the identity into the original equation.

2*cos^2(x)−1−cos(x)=0

3. Rearrange the terms into a standard quadratic form.

2*cos^2(x)−cos(x)−1=0

4. Factor the quadratic expression by treating cos(x) as the variable.

(2*cos(x)+1)*(cos(x)−1)=0

5. Set each factor to zero to find the values of cos(x) that satisfy the identity.

2*cos(x)+1=0⇒cos(x)=−1/2

cos(x)−1=0⇒cos(x)=1

6. Solve for x within the standard interval [0,2*π)

x=(2*π)/3,(4*π)/3,0

Final Answer

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

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