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Using the property $ln(a)+ln(b) = ln(ab)$, we can rewrite the expression as $e^x + ln(sec^2(x)cos^2(x))$ which can then simplify to $e^x + ln(\frac{(cos^2x)}{(cos^2x)})$ since $sec(x) = \frac{1}{cos(x)}$.

It looks like you are working through the simplification of the expression:

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

Let's go step-by-step to simplify it into a single term.

### Step 1: Combine the logarithmic terms
We use the logarithmic identity you mentioned:
\[
\ln(a) + \ln(b) = \ln(a \cdot b).
\]
In this case, the two logarithmic terms are $\ln(\cos^2(x))$ and $\ln(1 + \tan^2(x))$, so:

\[
\ln(\cos^2(x)) + \ln(1 + \tan^2(x)) = \ln(\cos^2(x) \cdot (1 + \tan^2(x))).
\]

### Step 2: Use a trigonometric identity
Next, we recall that:
\[
1 + \tan^2(x) = \sec^2(x).
\]
Thus, the logarithmic expression becomes:
\[
\ln(\cos^2(x) \cdot \sec^2(x)).
\]

### Step 3: Simplify the trigonometric expression
Since $\sec(x) = \frac{1}{\cos(x)}$, we simplify $\cos^2(x) \cdot \sec^2(x)$ as follows:
\[
\cos^2(x) \cdot \sec^2(x) = \cos^2(x) \cdot \left(\frac{1}{\cos^2(x)}\right) = 1.
\]

Therefore, the logarithmic part simplifies to:
\[
\ln(1).
\]

### Step 4: Simplify the entire expression
Since $\ln(1) = 0$, the expression simplifies to:
\[
e^x - 0 = e^x.
\]

### Final Answer:
The simplified expression is:
\[
e^x.
\]

Let me know if you'd like further details or clarifications!

---

**5 related questions:**
1. How does the identity $ 1 + \tan^2(x) = \sec^2(x) $ arise from the Pythagorean identities?
2. What is the derivative of $ e^x $ and why is it unique?
3. How do logarithmic rules like $ \ln(a) + \ln(b) = \ln(ab) $ help in simplifying expressions?
4. Can you explain the connection between trigonometric identities and exponential functions?
5. How does the natural logarithm $ \ln(x) $ behave when $ x = 1 $?

**Tip:** Always check if trigonometric identities can simplify logarithmic expressions, as these often lead to significant reductions in complexity.

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

Simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic identities and trigonometric simplifications. This problem involves key concepts such as logarithms, exponential functions, and trigonometric identities, suitable for Grades 10-12.

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