Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0,2π) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate "No Solution."

cos2(x)+5cos(x)=2cos(x)−2

Let's solve the given trigonometric equation:

\[
\cos^2(x) + 5\cos(x) = 2\cos(x) - 2
\]

### Step 1: Move all terms to one side of the equation

We begin by rearranging the terms so that all terms are on one side of the equation. First, subtract \(2\cos(x) - 2\) from both sides:

\[
\cos^2(x) + 5\cos(x) - 2\cos(x) + 2 = 0
\]

Simplify the equation:

\[
\cos^2(x) + 3\cos(x) + 2 = 0
\]

### Step 2: Use substitution

Let \( y = \cos(x) \). This simplifies the equation into a quadratic form:

\[
y^2 + 3y + 2 = 0
\]

### Step 3: Solve the quadratic equation

Now, solve the quadratic equation \( y^2 + 3y + 2 = 0 \) by factoring:

\[
(y + 1)(y + 2) = 0
\]

So, the solutions for \( y \) are:

\[
y = -1 \quad \text{or} \quad y = -2
\]

### Step 4: Solve for \( x \)

Recall that \( y = \cos(x) \), so we now solve for \( x \):

1. For \( \cos(x) = -1 \):

\[
x = \pi + 2n\pi \quad \text{where} \, n \, \text{is any integer.}
\]

In the interval \( [0, 2\pi) \), the solution is:

\[
x = \pi
\]

2. For \( \cos(x) = -2 \):

Since the cosine function only takes values between \(-1\) and \(1\), there is no solution for \( \cos(x) = -2 \).

### Final Solutions:

The only solution in the interval \( [0, 2\pi) \) is:

\[
x = \pi
\]

For the general solution:

\[
x = \pi + 2n\pi \quad \text{where} \, n \, \text{is any integer.}
\]

### Summary:
- In the interval \( [0, 2\pi) \), the solution is \( x = \pi \).
- The general solution is \( x = \pi + 2n\pi \), where \( n \) is any integer.

Would you like more details or have any questions?

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**Related questions:**
1. How would the solution change if the equation involved sine instead of cosine?
2. What are the trigonometric identities related to the double-angle formulas for cosine?
3. How can you solve \( \cos^2(x) - 2\cos(x) + 1 = 0 \)?
4. Can you derive the general solution for \( \sin(x) = -1 \)?
5. How do you solve the equation \( \tan^2(x) - 3\tan(x) + 2 = 0 \)?

**Tip:** When solving trigonometric equations, try using substitution to turn complex trig functions into simpler algebraic forms.

Use trigonometric identities and algebraic methods, as necessary, to solve the following trigonometric equation. Please identify all possible solutions by including all answers in [0,2π) and indicating the remaining answers by using n to represent any integer. Round your answer to four decimal places, if necessary. If there is no solution, indicate 'No Solution.'

cos^2(x) + 5cos(x) = 2cos(x) − 2.

Learn how to solve the trigonometric equation cos^2(x) + 5cos(x) = 2cos(x) - 2 step by step. This detailed guide explores key trigonometric and algebraic concepts, including quadratic equations and cosine properties. Suitable for students in Grades 10-12, with all solutions in [0, 2π) and general solutions included.

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