We are given the expression:

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

We will simplify it step by step.

Step 1: Simplify the logarithmic terms

Use the logarithmic identity for powers:

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

So the expression becomes:

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

Step 2: Use the identity for $\tan^2(x)$

Recall that:

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

Thus, the expression now is:

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

Step 3: Simplify $\ln(\sec^2(x))$

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

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

Using the previous result $\ln(\cos^2(x)) = 2 \ln(\cos(x))$, we get:

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

Step 4: Combine the terms

Now substitute this back into the expression:

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

Simplify:

$e^x - 4 \ln(\cos(x))$

This is the simplified expression. There is no further simplification possible without making assumptions about the value of $x$.

Final simplified expression:

$e^x - 4 \ln(\cos(x))$

Would you like more details or clarifications on any step?

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Here are some related questions you can explore:

1. What are the properties of the exponential function $e^x$?
2. How do logarithmic identities help in simplifying expressions?
3. What is the relationship between $\sec(x)$ and $\cos(x)$?
4. How can we apply this expression for specific values of $x$?
5. Can this expression be simplified further for small values of $x$?

Tip: When simplifying logarithms, always look for power or quotient rules to break down complicated terms easily!