You are given the expression:

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

### Step 1: Simplify the logarithmic terms
Notice that \(\ln(\cos^2(x))\) is repeated twice. So, let's combine those terms:

\[
\ln(\cos^2(x)) + \ln(\cos^2(x)) = 2\ln(\cos^2(x))
\]

### Step 2: Apply logarithmic properties
Using the logarithmic identity \( \ln(a^b) = b\ln(a) \), we can simplify \( \ln(\cos^2(x)) \) further:

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

Thus, the expression becomes:

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

### Final simplified expression:
\[
e^x - 4\ln(\cos(x))
\]

This is the simplified version of the original expression.

Would you like more details, or do you have any questions on this? Here are some related questions to explore further:

1. What is the derivative of \(e^x - 4\ln(\cos(x))\)?
2. How does the domain of the expression change after simplification?
3. Can we evaluate this expression for specific values of \(x\)?
4. What happens to the expression when \(x = 0\)?
5. How does this expression behave as \(x\) approaches infinity?

**Tip:** When dealing with logarithms of trigonometric functions, remember to check the domain restrictions to ensure the function is defined.

Discover how to simplify the expression e^x - ln(cos^2(x)) - ln(cos^2(x)) by applying logarithmic properties and trigonometric identities. This detailed step-by-step solution explains key concepts from logarithms and exponential functions, suitable for Grades 11-12 students.

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