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!