Solve for x 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) only.

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, treating cos(x) as the variable.

(2*cos(x)+1)*(cos(x)−1)=0

5. 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

6. Solve for x using the unit circle or inverse trigonometric properties, including the general solution where n is an integer.

cos(x)=1⇒x=2*n*π

cos(x)=−1/2⇒x=(2*π)/3+2*n*π

cos(x)=−1/2⇒x=(4*π)/3+2*n*π

Final Answer

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

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