To simplify the expression \( e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x)) \), let’s break it down into components and evaluate each term.

### Step 1: Simplify \( \ln(\cos^2(x)) \)
Using the logarithmic identity \( \ln(a^b) = b \ln(a) \), we can rewrite \( \ln(\cos^2(x)) \) as:
\[
\ln(\cos^2(x)) = 2 \ln(\cos(x))
\]

### Step 2: Simplify \( \ln(1 + \tan^2(x)) \)
Recall that \( 1 + \tan^2(x) = \sec^2(x) \) (using a trigonometric identity). Therefore, we can rewrite \( \ln(1 + \tan^2(x)) \) as:
\[
\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))
\]

### Step 3: Substitute and Simplify the Expression
Now, substitute these simplified forms back into the original expression:
\[
e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x)) = e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x))
\]

Since \( \sec(x) = \frac{1}{\cos(x)} \), \( \ln(\sec(x)) = -\ln(\cos(x)) \). Substituting this:
\[
e^x - 2 \ln(\cos(x)) + 2(-\ln(\cos(x))) = e^x
\]

Thus, the simplified form of the expression is:
\[
\boxed{e^x}
\]

Would you like further clarification on any steps, or do you have questions?

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Here are some related questions that might expand your understanding:
1. How would this simplification change if we added \( \sin(x) \) to the expression?
2. What happens if we replace \( \tan(x) \) with \( \cot(x) \) in a similar expression?
3. How can we simplify an expression like \( e^{x+y} - \ln(\sin^2(x) + \cos^2(x)) \)?
4. What are common properties of logarithmic and exponential functions that help in simplification?
5. How would this change if we worked in radians versus degrees for the trigonometric functions?

**Tip:** Remember that logarithmic identities are particularly useful for simplifying expressions involving powers and products.

This guide provides a step-by-step simplification of the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)), using logarithmic properties and trigonometric identities. Ideal for high school students studying advanced functions, the solution breaks down each term for clarity.

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