To solve the equation \(\tan(x) = 1\), we need to determine the values of \(x\) for which the tangent function equals 1.

### Analyzing the Tangent Function

The tangent function, \(\tan(x)\), has a period of \(\pi\) because it repeats every \(\pi\) radians. The function \(\tan(x) = 1\) at the angles where the sine and cosine of \(x\) are equal, given that \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).

### Finding the Solutions

The tangent function equals 1 at:

\[
x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

where \(n\) is any integer. This is because \(\tan\left(\frac{\pi}{4}\right) = 1\) and, due to the periodic nature of the tangent function, this result repeats every \(\pi\) radians.

### General Solution

The general solution for \(x\) when \(\tan(x) = 1\) is:

\[
x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z}
\]

This means that the specific values of \(x\) can be \(\frac{\pi}{4}\), \(\frac{5\pi}{4}\), \(\frac{9\pi}{4}\), and so on for positive \(n\), or \(-\frac{3\pi}{4}\), \(-\frac{7\pi}{4}\), etc., for negative \(n\).

Would you like further details on this, or do you have any questions?

Here are some related questions to consider:
1. How do we solve trigonometric equations for other values, such as \(\tan(x) = -1\)?
2. What are the properties of the tangent function, and how does it differ from sine and cosine?
3. How can we solve trigonometric equations that involve multiple angles, like \(\tan(2x) = 1\)?
4. What are the implications of the period of the tangent function on solving equations?
5. How would we graphically interpret the solutions to \(\tan(x) = 1\)?

**Tip:** Knowing the general form of solutions for trigonometric equations can help you quickly find all possible solutions over any interval.

Explore how to solve the equation tan(x) = 1 with detailed trigonometric analysis. Learn about the properties of the tangent function and find specific solutions. Suitable for advanced high school students and beyond.

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