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To simplify the expression given:

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

we can break it down step-by-step and use properties of logarithms and trigonometric identities.

### Step 1: Simplify \(\ln(\cos^2(x))\)
Using the logarithmic property \(\ln(a^b) = b \cdot \ln(a)\), we get:
\[
\ln(\cos^2(x)) = 2 \ln(\cos(x))
\]

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

### Step 2: Simplify \(\ln(1 + \tan^2(x))\)
Using the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\), we replace \(1 + \tan^2(x)\) with \(\sec^2(x)\):
\[
\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))
\]

### Step 3: Substitute and Combine Terms
Now the expression is:
\[
e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x))
\]

Using \(\sec(x) = \frac{1}{\cos(x)}\), we have:
\[
\ln(\sec(x)) = -\ln(\cos(x))
\]
Thus, \(2 \ln(\sec(x)) = -2 \ln(\cos(x))\).

Substituting this back in, we get:
\[
e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x)) = e^x
\]

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

Would you like further clarification on any of the steps?

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### Additional Questions
1. How would this simplification change if the expression had a different trigonometric function?
2. What happens if we replace \(\cos^2(x)\) with \(\sin^2(x)\) in the expression?
3. Can you simplify the expression if it included a different logarithmic function base, like \(\log_{10}\)?
4. What other trigonometric identities can help simplify similar expressions?
5. How would the approach differ if \(\tan(x)\) was replaced by \(\cot(x)\)?

### Tip
Remember that recognizing trigonometric identities, such as \(1 + \tan^2(x) = \sec^2(x)\), can significantly simplify complex expressions.

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

Explore how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using properties of logarithms and trigonometric identities. This problem is ideal for undergraduate students and covers exponential functions, logarithmic properties, and trigonometric identities.

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