The equation given is:

\[
\cos^2(x) + \sin(x) = 1
\]

We can solve this step by step.

### Step 1: Use the Pythagorean identity.
Recall the identity:

\[
\cos^2(x) = 1 - \sin^2(x)
\]

However, in this equation, it is better to keep \(\cos^2(x)\) as it is and work with \(\sin(x)\).

### Step 2: Rearrange the equation.
Start by rewriting the equation:

\[
\cos^2(x) = 1 - \sin(x)
\]

### Step 3: Use the identity \(\cos^2(x) = 1 - \sin^2(x)\).
Now, substitute \(1 - \sin^2(x)\) in place of \(\cos^2(x)\):

\[
1 - \sin^2(x) = 1 - \sin(x)
\]

### Step 4: Simplify the equation.
Now, subtract 1 from both sides:

\[
-\sin^2(x) = -\sin(x)
\]

Multiplying both sides by \(-1\) gives:

\[
\sin^2(x) = \sin(x)
\]

### Step 5: Solve the quadratic equation.
Rearrange the equation:

\[
\sin^2(x) - \sin(x) = 0
\]

Factor the equation:

\[
\sin(x)(\sin(x) - 1) = 0
\]

### Step 6: Solve for \(\sin(x)\).
From the factored equation, we get two solutions:

1. \(\sin(x) = 0\)
2. \(\sin(x) = 1\)

### Step 7: Find the general solutions.

- If \(\sin(x) = 0\), then:
  \[
  x = n\pi, \quad n \in \mathbb{Z}
  \]
  
- If \(\sin(x) = 1\), then:
  \[
  x = \frac{\pi}{2} + 2n\pi, \quad n \in \mathbb{Z}
  \]

### Final Solutions:
Thus, the general solutions are:

\[
x = n\pi \quad \text{or} \quad x = \frac{\pi}{2} + 2n\pi, \quad n \in \mathbb{Z}
\]

Would you like more details or further clarification?

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Here are five related questions to consider:

1. What is the general solution for \(\cos(x) = 0\)?
2. How would the solution change if the equation was \(\cos^2(x) + \sin(x) = 2\)?
3. What is the Pythagorean identity, and how is it used in solving trigonometric equations?
4. What are the periodic properties of sine and cosine functions?
5. Can you explain how to factor quadratic trigonometric equations?

**Tip**: Always look for opportunities to apply known identities, like Pythagorean identities, in trigonometric equations.

Learn how to solve the trigonometric equation cos^2(x) + sin(x) = 1 step by step using trigonometric identities and quadratic factoring. Explore key concepts in trigonometry such as the Pythagorean identity and solve for general solutions. Suitable for students in Grades 10-12.

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