cos2x-cosx+1=0

solve this

To solve the equation cos(2x) - cos(x) + 1 = 0, we can use some trigonometric identities and algebraic techniques. Here's a step-by-step explanation:

Step 1: Rewrite cos(2x) using the double-angle formula.
cos(2x) = 2cos^2(x) - 1

Substituting this into the equation gives:
2cos^2(x) - 1 - cos(x) + 1 = 0

Step 2: Combine like terms.
2cos^2(x) - cos(x) = 0

Step 3: Factor out common terms.
cos(x)(2cos(x) - 1) = 0

Now we have two possibilities:

Case 1: cos(x) = 0
If cos(x) = 0, then x could be any angle that satisfies cos(x) = 0. These angles are π/2 + πn, where n is an integer.

Case 2: 2cos(x) - 1 = 0
Solving this equation, we get:
2cos(x) = 1
cos(x) = 1/2

To find the angles that satisfy cos(x) = 1/2, we can use the unit circle or inverse trigonometric functions. The solutions for cos(x) = 1/2 are x = π/3 + 2πn and x = 5π/3 + 2πn, where n is an integer.

So, the complete solution is:
x = π/2 + πn, π/3 + 2πn, 5π/3 + 2πn

where n is an integer.