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The equation provided is:

\[
\cos(\theta) \cdot (\cos(\theta) - 2) = 1 \quad \text{for} \quad -\pi \leq \theta \leq \pi.
\]

Let's solve this equation step by step.

### Step 1: Expand the equation
First, expand the left-hand side:

\[
\cos^2(\theta) - 2\cos(\theta) = 1.
\]

### Step 2: Rearrange the equation
Now, let's rearrange the equation to set it equal to zero:

\[
\cos^2(\theta) - 2\cos(\theta) - 1 = 0.
\]

### Step 3: Solve the quadratic equation
This is a quadratic equation in terms of \(\cos(\theta)\). Let \(x = \cos(\theta)\). The equation becomes:

\[
x^2 - 2x - 1 = 0.
\]

We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a},
\]

where \(a = 1\), \(b = -2\), and \(c = -1\).

### Step 4: Apply the quadratic formula
Substituting the values into the formula:

\[
x = \frac{2 \pm \sqrt{(-2)^2 - 4(1)(-1)}}{2(1)} = \frac{2 \pm \sqrt{4 + 4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{2 \pm 2\sqrt{2}}{2} = 1 \pm \sqrt{2}.
\]

Thus, the possible values for \(x = \cos(\theta)\) are:

\[
\cos(\theta) = 1 + \sqrt{2} \quad \text{or} \quad \cos(\theta) = 1 - \sqrt{2}.
\]

### Step 5: Analyze the solutions
However, \(\cos(\theta)\) must lie within the range \([-1, 1]\). 

- \(1 + \sqrt{2}\) is greater than 1, so it is not a valid solution.
- \(1 - \sqrt{2}\) is within the valid range, and thus:

\[
\cos(\theta) = 1 - \sqrt{2}.
\]

### Step 6: Determine the corresponding values of \(\theta\)
Finally, find the corresponding values of \(\theta\) for \(\cos(\theta) = 1 - \sqrt{2}\). This value corresponds to specific angles within the interval \([- \pi, \pi]\). Given that \(1 - \sqrt{2}\) is a specific value, \(\theta\) can be found using:

\[
\theta = \cos^{-1}(1 - \sqrt{2}).
\]

However, this value is not among common angles and would likely require numerical methods to find an exact value or to approximate.

### Conclusion
The solution for \(\theta\) within the interval \([- \pi, \pi]\) corresponds to the angle where \(\cos(\theta) = 1 - \sqrt{2}\).

Would you like further details on how to compute this angle or explore related concepts?

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**Relative Questions:**

1. What are the properties of the cosine function over the interval \([- \pi, \pi]\)?
2. How would the quadratic formula apply if the coefficients were different?
3. Can \(\cos^{-1}(1 - \sqrt{2})\) be represented in a simpler form?
4. How would the solution change if the equation involved \(\sin(\theta)\) instead of \(\cos(\theta)\)?
5. What are the implications of having a non-real solution in trigonometric equations?

**Tip:** When solving trigonometric equations involving quadratic forms, always check if the roots fall within the valid range of the trigonometric function.

Solve the equation: cos(θ)(cos(θ) - 2) = 1 for -π ≤ θ ≤ π.

Learn how to solve a trigonometric quadratic equation involving the cosine function: cos(θ)(cos(θ) - 2) = 1. This step-by-step guide covers the quadratic formula and cosine function properties, tailored for high school students in Grades 11-12.

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