The given expression is:

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

Let's simplify it step by step:

Step 1: Simplifying the logarithmic terms

1. The second term: $\ln(\cos^2(x)) = 2 \ln(\cos(x))$ Using the logarithmic property $\ln(a^b) = b \ln(a)$.

2. The third term: $\ln(1 + \tan^2(x))$ Here, we use the identity $1 + \tan^2(x) = \sec^2(x)$. Therefore: $\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))$

Step 2: Substitute and simplify

Now, substituting these into the original expression:

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

Since $\ln(\sec(x)) = -\ln(\cos(x))$, we can replace $2 \ln(\sec(x))$ with $-2 \ln(\cos(x))$:

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

Thus, the expression simplifies further:

$e^x$

Final Answer:

The entire expression simplifies to $e^x$.

---

Do you need further details or have any questions?

Here are 5 related questions you might find interesting:

1. How can we derive other trigonometric identities involving logarithms?
2. Can we generalize this result to other exponential and logarithmic combinations?
3. What is the derivative of $e^x$?
4. How would the result change if $\cos^2(x)$ were replaced by $\sin^2(x)$?
5. What are the properties of the natural logarithm that help simplify such expressions?

Tip: When simplifying logarithmic expressions, always check for possible trigonometric identities to make simplifications easier.