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The given differential equation is:

\[
\frac{dy}{dx} = \tan^2(x)
\]

### Solution:
We can solve this by separating the variables and integrating.

1. **Separate the variables**:
\[
dy = \tan^2(x) \, dx
\]

2. **Use trigonometric identity**: 
Recall that:
\[
\tan^2(x) = \sec^2(x) - 1
\]
So, the equation becomes:
\[
dy = (\sec^2(x) - 1) \, dx
\]

3. **Integrate both sides**:
\[
y = \int (\sec^2(x) - 1) \, dx
\]

Now, integrate term by term:

\[
y = \int \sec^2(x) \, dx - \int 1 \, dx
\]

The integrals are well-known:
\[
\int \sec^2(x) \, dx = \tan(x)
\]
\[
\int 1 \, dx = x
\]

Thus:
\[
y = \tan(x) - x + C
\]
where \(C\) is the constant of integration.

### Final solution:
\[
y = \tan(x) - x + C
\]

Would you like further details on the steps or have any questions?

### Related Questions:
1. What is the general method for solving separable differential equations?
2. How does the trigonometric identity for \(\tan^2(x)\) help in solving differential equations?
3. Can you explain why we introduce a constant of integration in indefinite integrals?
4. How would you solve the equation if an initial condition were provided, like \( y(0) = 0 \)?
5. What are some other common integrals involving trigonometric functions?

### Tip:
Always check if a trigonometric identity can simplify the equation before integrating, as this often makes the process much easier.

Solve the following differential equation: dy/dx = tan^2(x)

Learn how to solve the differential equation dy/dx = tan^2(x) using integration and trigonometric identities. This problem involves separating variables, applying the identity tan^2(x) = sec^2(x) - 1, and solving through integration. Suitable for high school students in Grades 11-12.

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