Solve for x 4cos(x)^2-4cos(x)+1=0

Problem

4*cos^2(x)−4*cos(x)+1=0

Solution

1. Identify the equation as a quadratic in terms of cos(x)

2. Substitute u=cos(x) to rewrite the equation as a standard quadratic.

4*u^2−4*u+1=0

3. Factor the quadratic expression, noting that it is a perfect square trinomial of the form (a*x−b)^2

(2*u−1)^2=0

4. Solve for u by taking the square root of both sides and isolating the variable.

2*u−1=0

u=1/2

5. Back-substitute cos(x) for u to find the values of x

cos(x)=1/2

6. Determine the general solution for x based on the unit circle, where the cosine is 1/2 at π/3 and −π/3 (or (5*π)/3.

x=π/3+2*n*π

x=−π/3+2*n*π

Final Answer

x=±π/3+2*n*π

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