The equation \( \sin(x) + \cos(x) = 1 \) can be solved by considering trigonometric identities and algebraic manipulations. Let's solve it step by step.

### Step 1: Square both sides
Square both sides of the equation to use the Pythagorean identity:

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

Expanding the left-hand side:

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

### Step 2: Use the Pythagorean identity
The Pythagorean identity tells us that \( \sin^2(x) + \cos^2(x) = 1 \). Substituting this into the equation:

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

### Step 3: Simplify the equation
Subtract 1 from both sides:

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

### Step 4: Factor the equation
The product \( 2\sin(x)\cos(x) \) equals zero if either \( \sin(x) = 0 \) or \( \cos(x) = 0 \).

#### Case 1: \( \sin(x) = 0 \)
The sine function equals zero at:

\[
x = n\pi, \quad \text{where } n \text{ is an integer}
\]

#### Case 2: \( \cos(x) = 0 \)
The cosine function equals zero at:

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

### Step 5: Check the original equation
We should check if these solutions satisfy the original equation \( \sin(x) + \cos(x) = 1 \).

1. **For \( x = n\pi \)**: 
   - \( \sin(x) = 0 \) and \( \cos(x) = (-1)^n \). 
   - Therefore, \( \sin(x) + \cos(x) = (-1)^n \), which equals 1 only when \( n \) is even (i.e., \( n = 2k \) where \( k \) is an integer).

2. **For \( x = \frac{\pi}{2} + n\pi \)**:
   - \( \sin(x) = (-1)^n \) and \( \cos(x) = 0 \).
   - Therefore, \( \sin(x) + \cos(x) = (-1)^n \), which equals 1 only when \( n \) is even.

Thus, the solution to \( \sin(x) + \cos(x) = 1 \) is:

\[
x = 2k\pi \quad \text{or} \quad x = \frac{\pi}{2} + 2k\pi, \quad \text{where } k \text{ is an integer}.
\]

Would you like more details or have any questions?

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Here are some related questions to deepen your understanding:

1. How does squaring both sides of an equation affect the solution set?
2. What are the main trigonometric identities, and how are they applied in problem-solving?
3. Can \( \sin(x) + \cos(x) = 1 \) be solved graphically? What would the graph look like?
4. How do phase shifts in trigonometric functions affect their solutions?
5. What are the periodic properties of sine and cosine functions?

**Tip:** Always remember to check your solutions in the original equation after manipulating or transforming it, especially when squaring both sides, as it may introduce extraneous solutions.

Learn how to solve the equation sin(x) + cos(x) = 1 using trigonometric identities and algebraic manipulations. This problem explores the application of the Pythagorean identity in trigonometry and is suitable for advanced high school students and beyond.

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