To solve the equation \( \sin(a) + \cos(a) = 1 \) and find the value of \( \tan(2a) \), follow these steps:

### Step 1: Express \( \sin(a) + \cos(a) \) in terms of trigonometric identities.
We know that:

\[
\sin(a) + \cos(a) = \sqrt{2} \sin\left(a + \frac{\pi}{4}\right)
\]

However, to solve the equation directly, we can square both sides:

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

Expanding the left side:

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

Using the Pythagorean identity \( \sin^2(a) + \cos^2(a) = 1 \), we get:

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

Subtract 1 from both sides:

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

### Step 2: Solve for \( \sin(2a) \).
Notice that \( \sin(2a) = 2\sin(a)\cos(a) \). So:

\[
\sin(2a) = 0
\]

This implies that:

\[
2a = n\pi \quad \text{for integer } n
\]

Thus, \( \sin(2a) = 0 \) when \( 2a \) is an integer multiple of \( \pi \).

### Step 3: Determine \( \tan(2a) \).
Since \( \sin(2a) = 0 \) and \( \tan(2a) = \frac{\sin(2a)}{\cos(2a)} \):

\[
\tan(2a) = \frac{0}{\cos(2a)} = 0
\]

### Final Answer:
\[
\tan(2a) = 0
\]

Would you like more details or have any further questions? Here are some related questions you might find interesting:

1. What are the values of \( a \) for which \( \sin(a) + \cos(a) = 1 \)?
2. How can you derive the identity \( \sin(a) + \cos(a) = \sqrt{2} \sin\left(a + \frac{\pi}{4}\right) \)?
3. How does the identity \( \tan(2a) = \frac{2\tan(a)}{1 - \tan^2(a)} \) relate to this problem?
4. What are the general solutions for \( \sin(2a) = 0 \)?
5. What is the geometric interpretation of the equation \( \sin(a) + \cos(a) = 1 \)?

**Tip:** Remember that \( \tan(2a) = 0 \) occurs when \( 2a = n\pi \) (where \( n \) is an integer), meaning \( a \) is a multiple of \( \frac{\pi}{2} \).

Discover how to solve the trigonometric equation where sin(a) + cos(a) = 1 and determine the value of tan(2a). This detailed step-by-step solution uses key trigonometric identities and is suitable for high school students in Grades 10-12.

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