We are given the expression:

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

Step-by-step simplification:

1. Simplify the logarithmic terms:

   The second term, $\ln(\cos^2(x))$, can be simplified using the logarithm property $\ln(a^b) = b\ln(a)$:

   $\ln(\cos^2(x)) = 2\ln(\cos(x))$

   So the expression becomes:

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

2. Recognize a trigonometric identity:

   The term $\ln(1 + \tan^2(x))$ can be simplified using the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$:

   $\ln(1 + \tan^2(x)) = \ln(\sec^2(x))$

   Since $\sec^2(x) = \frac{1}{\cos^2(x)}$, we now have:

   $\ln(\sec^2(x)) = 2\ln(\sec(x))$

   The expression becomes:

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

3. Use the relationship between $\sec(x)$ and $\cos(x)$:

   Recall that $\sec(x) = \frac{1}{\cos(x)}$, so $\ln(\sec(x)) = -\ln(\cos(x))$. Therefore, the term $2\ln(\sec(x))$ becomes:

   $2\ln(\sec(x)) = -2\ln(\cos(x))$

   Now, the expression is:

   $e^x - 2\ln(\cos(x)) - 2\ln(\cos(x))$

   Which simplifies to:

   $e^x$

Final simplified expression:

$e^x$

Would you like further details on any step, or have any questions?

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Here are 5 related questions for deeper understanding:

1. How does the identity $1 + \tan^2(x) = \sec^2(x)$ arise from trigonometric relationships?
2. What are the common logarithmic properties used in algebraic manipulations?
3. How can $\ln(a^b) = b\ln(a)$ be applied in other expressions involving logarithms?
4. Why does the term $2\ln(\sec(x))$ simplify to $-2\ln(\cos(x))$?
5. What are other ways to simplify logarithmic and exponential expressions?

Tip: Always look for fundamental identities or properties (like logarithms and trigonometric identities) to simplify complex expressions!