Let's simplify the expression step by step:

Given Expression:

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

Step 1: Simplify $\ln(\cos^2(x))$

We know the property of logarithms:

$\ln(a^b) = b \ln(a)$

Applying this to $\ln(\cos^2(x))$:

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

Step 2: Simplify $\ln(1 + \tan^2(x))$

Using the trigonometric identity:

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

Thus:

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

Again, using the property of logarithms:

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

Step 3: Combine the expression

Now, substitute these simplified forms back into the original expression:

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

Step 4: Simplify $\ln(\sec(x))$

Recall that:

$\sec(x) = \frac{1}{\cos(x)}$

So:

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

Step 5: Final Expression

Substituting $\ln(\sec(x)) = -\ln(\cos(x))$ back into the expression:

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

Simplify further:

$e^x - 2\ln(\cos(x)) - 2\ln(\cos(x)) = e^x - 4\ln(\cos(x))$

Final Simplified Form:

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

This is the fully simplified form of the given expression.

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Would you like further explanation, or do you have any questions?

Here are some related questions you might want to explore:

1. What is the derivative of this expression?
2. Can you integrate this expression with respect to $x$?
3. How would this expression change if we used different trigonometric identities?
4. Can this expression be further simplified under specific conditions for $x$?
5. What happens when $x = 0$ in the simplified expression?

Tip: When simplifying logarithmic expressions, always look for opportunities to apply logarithmic properties and trigonometric identities!