Let's walk through the solution step by step to simplify the given expression $e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))$.

Step 1: Simplifying $\ln(\cos^2(x))$

Using the logarithmic property $\ln(a^b) = b \ln(a)$, we have: $\ln(\cos^2(x)) = 2 \ln(\cos(x)).$ So the expression becomes: $e^x - 2 \ln(\cos(x)) + \ln(1 + \tan^2(x)).$

Step 2: Simplifying $\ln(1 + \tan^2(x))$

Using the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$, we get: $\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x)).$ Now the expression is: $e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x)).$

Step 3: Simplifying $\ln(\sec(x))$

Since $\sec(x) = \frac{1}{\cos(x)}$, we can rewrite $\ln(\sec(x))$ as: $\ln(\sec(x)) = \ln\left(\frac{1}{\cos(x)}\right) = - \ln(\cos(x)).$ Substitute this into the expression: $e^x - 2 \ln(\cos(x)) + 2 (- \ln(\cos(x))).$ This simplifies further to: $e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x)) = e^x - 4 \ln(\cos(x)).$

Final Simplified Expression:

Thus, the final simplified expression is: $e^x - 4 \ln(\cos(x)).$

Would you like further details on any of the steps?

Here are 5 related questions to explore further:

1. What are some properties of logarithms that are useful in simplifying expressions?
2. How does the identity $1 + \tan^2(x) = \sec^2(x)$ arise from the Pythagorean identities?
3. Can this expression $e^x - 4 \ln(\cos(x))$ be differentiated? What would its derivative be?
4. How would the expression change if $\cos^2(x)$ were replaced by $\sin^2(x)$?
5. What happens to the expression when $x = 0$?

Tip: Always keep logarithmic properties handy—they are essential for simplifying complex expressions.