To solve the equation \(2 = 2\tan(x)\), we can start by simplifying it:

\[
2 = 2\tan(x)
\]

Divide both sides by 2:

\[
1 = \tan(x)
\]

Now, we need to find the values of \(x\) for which \(\tan(x) = 1\).

The tangent function equals 1 at the following angles:

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

where \(n\) is any integer. This is because the tangent function has a period of \(\pi\), meaning it repeats every \(\pi\) radians.

### Solution

The general solution for \(x\) is:

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

This means that \(x\) can be \(\frac{\pi}{4}\), \(\frac{5\pi}{4}\), \(\frac{9\pi}{4}\), etc., or \(-\frac{3\pi}{4}\), \(-\frac{7\pi}{4}\), etc.

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

Here are some related questions to explore further:
1. How do we solve equations involving other trigonometric functions, like sine or cosine?
2. What are the key properties and graphs of the tangent function?
3. How does the period of a trigonometric function affect its general solution?
4. How would the solution change if the equation were \(2 = 2\tan(x) + 1\)?
5. What methods can be used to solve trigonometric equations with multiple angles, like \(\tan(2x) = 1\)?

**Tip:** Remember that for any trigonometric function, understanding its period and symmetry can help in finding all possible solutions.

Learn how to solve the equation 2 = 2tan(x) step-by-step. Understand the solutions involving the tangent function and its periodic properties. Suitable for advanced high school students and those studying trigonometry.

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