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The expression in the prompt asks you to simplify:

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

Let's go step by step.

### Step 1: Simplifying logarithmic terms
1. **\(\ln(\cos^2(x))\):**
   Using the logarithmic identity \(\ln(a^b) = b \ln(a)\), this simplifies to:
   \[
   \ln(\cos^2(x)) = 2 \ln(\cos(x))
   \]
   So, the term becomes \(2 \ln(\cos(x))\).

2. **\(\ln(1 + \tan^2(x))\):**
   Recall that \(1 + \tan^2(x) = \sec^2(x)\) by the Pythagorean identity. Therefore:
   \[
   \ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))
   \]
   Since \(\ln(\sec(x)) = -\ln(\cos(x))\) (because \(\sec(x) = 1/\cos(x)\)), this becomes:
   \[
   2 \ln(\sec(x)) = -2 \ln(\cos(x))
   \]

### Step 2: Combining the logarithmic terms
Now you have:

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

Therefore, after simplification, the expression reduces to:

\[
e^x
\]

The expression simplifies into just \(e^x\).

---

Do you have any questions or would you like further details on any part of the process?

### Relative Questions:
1. What other logarithmic identities can be used to simplify expressions like this?
2. How does the identity \(\ln(a^b) = b \ln(a)\) work with negative exponents?
3. Can you explain more about the Pythagorean identity used for \(\sec^2(x)\)?
4. Why does \(\ln(\sec(x)) = -\ln(\cos(x))\)?
5. How would the simplification change if the original expression had a different trigonometric function, like \(\sin(x)\)?

### Tip:
When simplifying logarithmic expressions, always look for trigonometric identities and logarithmic properties to simplify terms efficiently.

Consider the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) and then simplify it into a single term

Learn how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic and trigonometric identities. This problem includes key concepts in logarithms, trigonometric functions, and exponential functions, suitable for Grades 10-12 students.

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