To simplify the expression

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

let's break it down and simplify each component.

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

Using the property \(\ln(a^b) = b \ln(a)\), we have:
\[
\ln(\cos^2(x)) = 2 \ln(\cos(x))
\]

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

We can use the trigonometric identity:
\[
1 + \tan^2(x) = \sec^2(x)
\]
Thus:
\[
\ln(1 + \tan^2(x)) = \ln(\sec^2(x))
\]
Applying \(\ln(a^b) = b \ln(a)\), we get:
\[
\ln(\sec^2(x)) = 2 \ln(\sec(x))
\]

### Step 3: Substitute and Simplify

Substitute these simplified forms back into the original expression:

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

Since \(\sec(x) = \frac{1}{\cos(x)}\), we have:
\[
\ln(\sec(x)) = -\ln(\cos(x))
\]

Thus:
\[
2 \ln(\sec(x)) = -2 \ln(\cos(x))
\]

Now the expression becomes:
\[
e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x)) = e^x
\]

### Final Answer:
\[
e^x
\]

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

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Here are some related questions to deepen understanding:

1. How does the property \(\ln(a^b) = b \ln(a)\) apply to logarithmic expressions?
2. What other identities involving \(\tan(x)\) and \(\sec(x)\) might simplify complex expressions?
3. What are some common exponential and logarithmic simplifications?
4. How would this expression change if \( \cos(x) \) were replaced with \( \sin(x) \)?
5. What are the general steps for simplifying logarithmic expressions with trigonometric functions?

**Tip:** Always check for trigonometric identities when simplifying expressions with trig functions—they often reveal simplifications that aren't immediately obvious.

Explore a detailed solution to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)). This solution covers key calculus and trigonometric concepts, applying properties of logarithms and trigonometric identities to reach the final answer e^x. Suitable for advanced high school students in Grades 11-12.

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