Solve for x cos(2x)=cos(x)

Problem

cos(2*x)=cos(x)

Solution

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

cos(2*x)=2*cos2(x)−1

2. Substitute the identity into the original equation.

2*cos2(x)−1=cos(x)

3. Rearrange the equation into a standard quadratic form by subtracting cos(x) from both sides.

2*cos2(x)−cos(x)−1=0

4. Factor the quadratic expression.

(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 and the general solution for cosine, where n is any integer.
   For cos(x)=1

x=2*n*π

For cos(x)=−1/2

x=(2*π)/3+2*n*π

x=(4*π)/3+2*n*π

Final Answer

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

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