To simplify the expression \( e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x)) \), we can use the identity \( 1 + \tan^2(x) = \sec^2(x) \).

Thus, we have:

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

Since \( \ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right) \):

\[
\ln(\sec^2(x)) = -\ln(\cos^2(x))
\]

Now the expression becomes:

\[
e^x - \ln(\cos^2(x)) - \ln(\cos^2(x)) = e^x - 2\ln(\cos^2(x))
\]

Thus, the simplified form is:

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

Would you like more details on any specific part of this simplification? Here are some questions to consider:

1. What are the properties of logarithms that are used in this simplification?
2. Can you explain why \( 1 + \tan^2(x) = \sec^2(x) \)?
3. How does this expression behave for different values of \( x \)?
4. Are there any specific applications for this expression in calculus or physics?
5. Would you like to see a graph of the simplified expression? 

**Tip:** Remember that \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \) is a useful property for simplifying logarithmic expressions.

Explore the simplification of the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)), utilizing trigonometric identities and logarithmic properties. This solution is ideal for high school students studying advanced mathematics.

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