To simplify the expression $e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))$, let's break it down step by step:

1. Using the trigonometric identity for tangent: $1 + \tan^2(x) = \sec^2(x)$ So the expression $\ln(1 + \tan^2(x))$ becomes: $\ln(\sec^2(x))$ Since $\ln(a^b) = b \ln(a)$, we can rewrite this as: $\ln(\sec^2(x)) = 2\ln(\sec(x))$

2. Simplifying $\ln(\cos^2(x))$: Similarly, apply the logarithmic property $\ln(a^b) = b\ln(a)$: $\ln(\cos^2(x)) = 2\ln(\cos(x))$

3. Substitute back into the original expression: The expression now becomes: $e^x - 2\ln(\cos(x)) + 2\ln(\sec(x))$

4. Simplifying $\ln(\sec(x))$: Recall that $\sec(x) = \frac{1}{\cos(x)}$, so: $\ln(\sec(x)) = \ln\left(\frac{1}{\cos(x)}\right) = -\ln(\cos(x))$ Therefore: $2\ln(\sec(x)) = -2\ln(\cos(x))$

5. Final simplification: Now substitute $2\ln(\sec(x)) = -2\ln(\cos(x))$ into the expression: $e^x - 2\ln(\cos(x)) + (-2\ln(\cos(x))) = e^x$

So, the simplified expression is:

$e^x$

Would you like further details or clarifications?

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Here are 5 questions related to this simplification:

1. How can you apply logarithmic properties in other complex expressions?
2. What other trigonometric identities can be useful in simplifying logarithmic expressions?
3. Why does the logarithm of a reciprocal lead to a negative value?
4. What is the significance of the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$?
5. How does this simplification work in real-life applications involving exponential and logarithmic functions?

Tip: Always check for trigonometric identities when simplifying expressions involving trigonometric functions. They can significantly simplify the process!