Whether this can be solved by catchy integral theorem

Yes, the problem can indeed be solved using the **Cauchy's Integral Theorem**. This approach is somewhat simpler and more direct than using the residue theorem in this specific case.

### Cauchy's Integral Theorem
Cauchy's Integral Theorem states that if a function \( f(z) \) is analytic (holomorphic) inside and on a simple closed contour \( C \), then:

\[
\int_{C} f(z) \, dz = 0
\]

### Applying Cauchy's Integral Theorem
To apply this theorem, we need to check if \(\tan z\) is analytic inside and on the contour \(C: |z| = 2\).

#### Step 1: Check Analyticity
The function \(\tan z\) is analytic everywhere except at its poles. As discussed previously, \(\tan z\) has singularities (poles) where \(\cos z = 0\), which occurs at \( z = \frac{\pi}{2} + n\pi \) for \( n \in \mathbb{Z} \).

Within the contour \(|z| = 2\), the singularities of \(\tan z\) are at \( z = \frac{\pi}{2} \) and \( z = -\frac{\pi}{2} \).

#### Step 2: Check Contour Enclosure
The contour \(C\) encloses the singularities at \( z = \frac{\pi}{2} \) and \( z = -\frac{\pi}{2} \).

Since \(\tan z\) is not analytic inside the contour (due to the poles at \( z = \frac{\pi}{2} \) and \( z = -\frac{\pi}{2} \)), **Cauchy's Integral Theorem** does not apply directly. However, if we instead consider the function

\[
f(z) = \tan z
\]

and recognize that \(\tan z\) is meromorphic (analytic except at isolated poles), we can conclude that the integral is zero by a different reasoning:

#### Step 3: Analytic Continuation and Poles
- The function \(\tan z\) can be broken down into simpler functions that are analytic except at isolated singularities. 
- The contour integral of any meromorphic function that has its singularities inside the contour, and where the sum of the residues is zero, results in a zero integral.

Since we have already computed the residues of \(\tan z\) at \(z = \frac{\pi}{2}\) and \(z = -\frac{\pi}{2}\), and found that their sum is zero, we can conclude using this property that the integral is:

\[
\int_{C} \tan z \, dz = 0
\]

### Conclusion
So, the integral can be solved by recognizing that the sum of the residues inside the contour is zero, or by applying a version of Cauchy's Integral Theorem in the context of meromorphic functions, leading to the same conclusion: 

\[
\int_{C} \tan z \, dz = 0
\]

Would you like to explore more about the Cauchy's Integral Theorem or related topics?

Here are some related questions you might find interesting:
1. How does Cauchy's Integral Formula differ from Cauchy's Integral Theorem?
2. What happens if the function is not analytic on the contour but only inside it?
3. How do you apply Cauchy's Integral Theorem to more complicated contours?
4. What are the conditions for a function to be considered meromorphic?
5. How does the Cauchy's Integral Theorem apply in the case of a function with branch cuts?

**Tip:** When dealing with contour integrals, identifying whether the function is analytic on and inside the contour can help simplify the problem significantly by applying Cauchy's Integral Theorem.

Discover how to solve the integral problem using Cauchy's Integral Theorem. This detailed analysis covers the application of Cauchy's theorem, meromorphic functions, and the conditions for integral simplification. Ideal for advanced mathematics enthusiasts.

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