Find the Roots (Zeros) cos(2x)-cos(x)=0

Problem

cos(2*x)−cos(x)=0

Solution

1. Apply the double-angle identity for cosine, cos(2*x)=2*cos2(x)−1 to rewrite the equation in terms of cos(x)

2*cos2(x)−1−cos(x)=0

2. Rearrange the terms into a standard quadratic form.

2*cos2(x)−cos(x)−1=0

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

(2*cos(x)+1)*(cos(x)−1)=0

4. Set each factor to zero to find the possible values for cos(x)

2*cos(x)+1=0⇒cos(x)=−1/2

cos(x)−1=0⇒cos(x)=1

5. Solve for x using the unit circle values and adding the period 2*π*n where n is an integer.

cos(x)=1⇒x=2*π*n

cos(x)=−1/2⇒x=(2*π)/3+2*π*n

cos(x)=−1/2⇒x=(4*π)/3+2*π*n

Final Answer

cos(2*x)−cos(x)=0⇒x=2*π*n,(2*π)/3+2*π*n,(4*π)/3+2*π*n

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