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The given expression is:

\[
e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))
\]

### Step-by-step simplification:

1. **Use a trigonometric identity**:
   Recall the identity for \( \tan^2(x) \):
   \[
   1 + \tan^2(x) = \sec^2(x)
   \]
   This allows us to replace \( \ln(1 + \tan^2(x)) \) with \( \ln(\sec^2(x)) \).

2. **Simplify the logarithmic terms**:
   The expression becomes:
   \[
   e^x - \ln(\cos^2(x)) + \ln(\sec^2(x))
   \]
   
   We also know that \( \sec(x) = \frac{1}{\cos(x)} \), so \( \sec^2(x) = \frac{1}{\cos^2(x)} \). Substituting this into the logarithmic term:
   \[
   e^x - \ln(\cos^2(x)) + \ln\left(\frac{1}{\cos^2(x)}\right)
   \]
   
3. **Combine the logarithms**:
   Using the property of logarithms \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \), the expression simplifies to:
   \[
   e^x - \ln(\cos^2(x)) + \left( - \ln(\cos^2(x)) \right)
   \]
   This simplifies further as the logarithmic terms cancel out:
   \[
   e^x
   \]

### Final Answer:
The simplified expression is:
\[
e^x
\]

Would you like more details or have any questions?

### Related questions:
1. Can you explain the logarithmic properties used in this simplification?
2. How does the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\) work?
3. Can you show a different way to simplify this expression?
4. What are some other important logarithmic rules to remember?
5. How do exponential functions behave in relation to logarithms?

### Tip:
Always remember to use trigonometric identities when simplifying expressions with logarithms and trigonometric functions!

Consider the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) and simplify it into a single term.

Learn how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) step by step. This problem explores the use of trigonometric identities and logarithmic properties, ideal for students in advanced high school calculus or undergraduate mathematics.

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