The true topics involved in this solution encompass:

1. Exponential and Logarithmic Functions: The expression includes the exponential function $e^x$ and the logarithmic function $\ln(x)$. You use logarithmic properties, such as $\ln(a^b) = b\ln(a)$, to simplify terms like $\ln(\cos^2(x))$.

2. Trigonometric Identities: The trigonometric identity $1 + \tan^2(x) = \sec^2(x)$ is applied to simplify $\ln(1 + \tan^2(x))$ into $\ln(\sec^2(x))$.

3. Logarithmic Manipulations: The properties of logarithms are used throughout, including the fact that $\ln(\frac{1}{a}) = -\ln(a)$. This is crucial when simplifying terms like $\ln(\sec(x))$ into $-\ln(\cos(x))$.

4. Trigonometric Function Relationships: The relationship between $\sec(x)$ and $\cos(x)$, i.e., $\sec(x) = \frac{1}{\cos(x)}$, is used to further simplify the expression.

5. Simplification Techniques: Algebraic simplifications, combining terms like $2\ln(\cos(x))$ and $2(-\ln(\cos(x)))$, are key to reaching the final expression.

These topics are foundational in calculus and pre-calculus, involving functions, identities, and the interplay between exponential, logarithmic, and trigonometric expressions.

Do you want details or have any questions?

Here are 5 related questions to deepen your understanding:

1. How does the property $\ln(a^b) = b\ln(a)$ apply to different functions?
2. What is the significance of the identity $1 + \tan^2(x) = \sec^2(x)$?
3. How does logarithmic simplification affect the domain of the functions involved?
4. What other trigonometric identities can simplify complex expressions like this?
5. How can this technique be used to solve logarithmic equations?

Tip: Always keep in mind domain restrictions when simplifying logarithmic and trigonometric functions, especially with $\ln(x)$ and $\cos(x)$.