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!