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The given problem is a first-order linear differential equation with an initial condition:

\[
y' \cdot \tan(x) - 3y = 3y \cdot \tan^2(x), \quad y\left(\frac{\pi}{4}\right) = 1.
\]

### Steps to solve:

1. **Rewrite the equation**:
   We can begin by rewriting the equation in a more standard form:
   
   \[
   y' \cdot \tan(x) = 3y \left( 1 + \tan^2(x) \right).
   \]

   Using the identity \( 1 + \tan^2(x) = \sec^2(x) \), the equation becomes:

   \[
   y' \cdot \tan(x) = 3y \cdot \sec^2(x).
   \]

2. **Divide through by \( \tan(x) \)** (assuming \( \tan(x) \neq 0 \)):

   \[
   y' = 3y \cdot \frac{\sec^2(x)}{\tan(x)}.
   \]

   This simplifies to:

   \[
   y' = 3y \cdot \frac{1}{\sin(x) \cdot \cos(x)} = 6y \cdot \csc(2x),
   \]
   using the identity \( \sin(2x) = 2 \sin(x) \cos(x) \), so \( \csc(2x) = \frac{1}{\sin(2x)} \).

3. **Solve the differential equation**:
   The equation is separable. We can write it as:

   \[
   \frac{dy}{y} = 6 \cdot \csc(2x) \, dx.
   \]

   Now integrate both sides:

   \[
   \int \frac{1}{y} \, dy = 6 \int \csc(2x) \, dx.
   \]

   The left-hand side integrates to \( \ln|y| \), and the right-hand side integrates to \( -3 \ln\left| \tan(x) \right| \), giving:

   \[
   \ln|y| = -3 \ln|\tan(x)| + C.
   \]

4. **Exponentiate both sides**:
   To solve for \( y \), exponentiate both sides:

   \[
   |y| = e^C \cdot |\tan(x)|^{-3}.
   \]

   Simplifying:

   \[
   y = A \cdot \tan^{-3}(x),
   \]

   where \( A = e^C \) is a constant.

5. **Apply the initial condition**:
   Using the initial condition \( y\left(\frac{\pi}{4}\right) = 1 \), substitute \( x = \frac{\pi}{4} \) and \( \tan\left(\frac{\pi}{4}\right) = 1 \):

   \[
   1 = A \cdot 1^{-3} = A.
   \]

   Therefore, \( A = 1 \), and the solution is:

   \[
   y = \frac{1}{\tan^3(x)}.
   \]

### Final solution:

\[
y(x) = \frac{1}{\tan^3(x)}.
\]

Would you like any further details or explanations?

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### Follow-up questions:
1. What are separable differential equations?
2. How can we verify if a differential equation is linear or nonlinear?
3. Can you explain the significance of the Cauchy problem in differential equations?
4. What are the key trigonometric identities used in solving such equations?
5. How would the solution change if the initial condition were different?

### Tip:
Always check whether the differential equation can be simplified using trigonometric identities before proceeding with solving it.

Solve the differential equation y' * tan(x) - 3y = 3y * tan^2(x), y(pi/4) = 1. (Cauchy problem)

Learn how to solve a first-order linear differential equation involving trigonometric identities. This problem demonstrates how to apply separation of variables, solve integrals, and use initial conditions. Suitable for undergraduate mathematics students.

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