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.