\(e^{x}-ln(cos^{2}(x))+ln(1+tan^{2}(x))\)e ^ x - ln(cos^2 (x)) + ln(1 + tan^2 (x))

We are tasked with simplifying the expression:

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

### Step-by-Step Simplification:

1. **Simplify \(\ln(1 + \tan^2(x))\):**
   Using the trigonometric identity:
   \[
   1 + \tan^2(x) = \sec^2(x)
   \]
   Thus, we can rewrite:
   \[
   \ln(1 + \tan^2(x)) = \ln(\sec^2(x))
   \]

2. **Apply logarithmic property:**
   Recall that \(\ln(a^b) = b \ln(a)\). Therefore:
   \[
   \ln(\sec^2(x)) = 2 \ln(\sec(x))
   \]
   
3. **Simplify \(\ln(\cos^2(x))\):**
   Using the same logarithmic property:
   \[
   \ln(\cos^2(x)) = 2 \ln(\cos(x))
   \]

4. **Substitute these into the original expression:**
   \[
   e^{x} - 2 \ln(\cos(x)) + 2 \ln(\sec(x))
   \]
   
5. **Simplify using \(\ln(\sec(x)) = -\ln(\cos(x))\):**
   \[
   e^{x} - 2 \ln(\cos(x)) + 2(-\ln(\cos(x))) = e^{x} - 2 \ln(\cos(x)) - 2 \ln(\cos(x))
   \]
   This simplifies to:
   \[
   e^{x}
   \]

Thus, the entire expression simplifies to:

\[
e^x
\]

Would you like to explore more details on the steps involved, or have any further questions?

### Related questions:
1. How does the trigonometric identity \(\sec(x) = \frac{1}{\cos(x)}\) work in simplifications?
2. What are the basic properties of logarithms that help in simplifying expressions?
3. How do you derive the identity \(1 + \tan^2(x) = \sec^2(x)\)?
4. Can you explain the derivative of \(e^x\) and its properties?
5. What are some applications of logarithmic and exponential functions in calculus?

**Tip:** Always check for trigonometric identities and logarithmic properties when simplifying expressions involving both!

Simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic properties and trigonometric identities. This step-by-step guide breaks down key concepts such as the logarithmic and trigonometric identities, making it suitable for high school students in Grades 11-12.

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