To simplify the expression \( e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x)) \), let’s break it down step-by-step:

### Step 1: Simplify the Logarithmic Terms

1. **Rewrite \( \ln(\cos^2(x)) \):**
   \[
   \ln(\cos^2(x)) = 2 \ln(\cos(x))
   \]

   So the expression becomes:
   \[
   e^x - 2 \ln(\cos(x)) + \ln(1 + \tan^2(x))
   \]

2. **Simplify \( \ln(1 + \tan^2(x)) \):**

   Recall that \( 1 + \tan^2(x) = \sec^2(x) \), so:
   \[
   \ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))
   \]

   Now, the expression is:
   \[
   e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x))
   \]

### Step 2: Substitute \( \ln(\sec(x)) = -\ln(\cos(x)) \)

   Since \( \sec(x) = \frac{1}{\cos(x)} \), we have:
   \[
   \ln(\sec(x)) = -\ln(\cos(x))
   \]

   Thus, \( 2 \ln(\sec(x)) = -2 \ln(\cos(x)) \), making the expression:
   \[
   e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x))
   \]

### Step 3: Combine Terms

   Combine \( -2 \ln(\cos(x)) - 2 \ln(\cos(x)) = 0 \), giving us:
   \[
   e^x
   \]

### Final Answer

\[
e^x
\]

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Would you like a more detailed breakdown or have any questions?

### Related Questions

1. How would this simplification change if \( e^x \) were a different exponential function?
2. How does this process differ if the base of the logarithm changes?
3. What are common simplifications involving \( \tan(x) \) and \( \sec(x) \)?
4. What is the effect of adding a constant term in logarithmic simplifications?
5. How would this simplification change in polar coordinates?

### Tip
When simplifying logarithmic expressions, look for identities like \( 1 + \tan^2(x) = \sec^2(x) \) to reduce complexity.

Explore how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) step by step, using logarithmic properties and trigonometric identities such as 1 + tan^2(x) = sec^2(x). This problem is suitable for students in Grades 11-12 who are learning about calculus and advanced algebra.

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