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The given equation is:

\[
2\sin 2x - 2\sin x + 2\cos x - 2 = 0
\]

### Step 1: Simplify the equation
We know the double-angle identity for sine:
\[
\sin 2x = 2\sin x \cos x
\]

Substitute \(\sin 2x = 2\sin x \cos x\) into the equation:

\[
2(2\sin x \cos x) - 2\sin x + 2\cos x - 2 = 0
\]

Simplify:
\[
4\sin x \cos x - 2\sin x + 2\cos x - 2 = 0
\]

### Step 2: Factorize
Group terms:
\[
2(2\sin x \cos x - \sin x + \cos x - 1) = 0
\]

Divide through by 2 (since \(2 \neq 0\)):
\[
2\sin x \cos x - \sin x + \cos x - 1 = 0
\]

### Step 3: Rearrange for factorization
Group terms involving \(\sin x\) and \(\cos x\):
\[
\sin x (2\cos x - 1) + (\cos x - 1) = 0
\]

Factorize:
\[
(\sin x + 1)(2\cos x - 1) = 0
\]

### Step 4: Solve for \(x\)
Set each factor to zero:

1. \(\sin x + 1 = 0\):
   \[
   \sin x = -1
   \]
   This occurs at:
   \[
   x = \frac{3\pi}{2} + 2n\pi \quad (n \in \mathbb{Z})
   \]

2. \(2\cos x - 1 = 0\):
   \[
   \cos x = \frac{1}{2}
   \]
   This occurs at:
   \[
   x = \pm \frac{\pi}{3} + 2n\pi \quad (n \in \mathbb{Z})
   \]

### Final Solution:
The solutions are:
\[
x = \frac{3\pi}{2} + 2n\pi \quad \text{and} \quad x = \pm \frac{\pi}{3} + 2n\pi \quad (n \in \mathbb{Z}).
\]

Would you like a deeper explanation of any step, or do you want me to explore specific cases? Here are five follow-up questions you might find helpful:

1. What are the steps to derive the double-angle identity used here?
2. How do we verify these solutions within the given equation?
3. Could we represent the solution graphically on a unit circle?
4. How do we solve trigonometric equations in general?
5. Can we rewrite the equation in terms of tangent or another trigonometric function?

**Tip:** Always consider trigonometric identities when simplifying equations—they're often the key to solving such problems efficiently!

Solve the equation: 2sin(2x) - 2sin(x) + 2cos(x) - 2 = 0.

Explore how to solve the trigonometric equation 2sin(2x) - 2sin(x) + 2cos(x) - 2 = 0 with a step-by-step solution. This problem uses key trigonometric identities, including the double-angle formula, to simplify and solve the equation. Suitable for students in Grades 11-12.

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