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

We are tasked with simplifying the expression:

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

### Step-by-Step Simplification

1. **Use logarithmic identities:**
   - Recall the logarithmic identity \( \ln(a^b) = b \cdot \ln(a) \).
   - Therefore, \( \ln(\cos^2(x)) = 2 \cdot \ln(\cos(x)) \).

   The expression now becomes:
   \[
   e^x - 2 \cdot \ln(\cos(x)) + \ln(1 + \tan^2(x))
   \]

2. **Substitute the trigonometric identity for \( \tan^2(x) \):**
   - Recall that \( 1 + \tan^2(x) = \sec^2(x) \).
   - Therefore, \( \ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \cdot \ln(\sec(x)) \).

   The expression is now:
   \[
   e^x - 2 \cdot \ln(\cos(x)) + 2 \cdot \ln(\sec(x))
   \]

3. **Simplify the logarithmic terms:**
   - Recall that \( \sec(x) = \frac{1}{\cos(x)} \), so \( \ln(\sec(x)) = -\ln(\cos(x)) \).
   - Now, substitute this into the expression:
   \[
   e^x - 2 \cdot \ln(\cos(x)) + 2 \cdot (-\ln(\cos(x)))
   \]
   - This simplifies to:
   \[
   e^x - 2 \cdot \ln(\cos(x)) - 2 \cdot \ln(\cos(x))
   \]

4. **Combine the logarithmic terms:**
   \[
   e^x - 4 \cdot \ln(\cos(x))
   \]

Thus, the simplified expression is:
\[
e^x - 4 \cdot \ln(\cos(x))
\]

Would you like further explanation or details on any of these steps?

### Related Questions:
1. What are the key properties of logarithmic functions?
2. How does the chain rule apply when differentiating \( e^x - 4 \cdot \ln(\cos(x)) \)?
3. What other trigonometric identities are commonly used in calculus?
4. How can we integrate \( e^x - 4 \cdot \ln(\cos(x)) \)?
5. What are the typical forms of simplifications using logarithmic identities?

**Tip:** Trigonometric identities and logarithmic properties often work together to simplify complex expressions!

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

This article provides a step-by-step solution to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)). It covers key mathematical concepts such as logarithmic identities, trigonometric identities, and properties of exponential functions. Suitable for advanced high school students in Grades 11-12.

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