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)

cos(2*x)=2*cos2(x)−1

2. Substitute the identity into the original equation to form a quadratic equation in terms of cos(x)

2*cos2(x)−1+cos(x)=0

3. Rearrange the terms into standard quadratic form.

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 by finding the angles that satisfy these values within the general solution.

cos(x)=1/2⇒x=π/3+2*n*π,x=(5*π)/3+2*n*π

cos(x)=−1⇒x=π+2*n*π

Final Answer

cos(2*x)+cos(x)=0⇒x=π*(2*n+1),x=±π/3+2*n*π

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