To simplify the expression given:

$e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))$

we can break it down step-by-step and use properties of logarithms and trigonometric identities.

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

Using the logarithmic property $\ln(a^b) = b \cdot \ln(a)$, we get: $\ln(\cos^2(x)) = 2 \ln(\cos(x))$

So, the expression becomes: $e^x - 2 \ln(\cos(x)) + \ln(1 + \tan^2(x))$

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

Using the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$, we replace $1 + \tan^2(x)$ with $\sec^2(x)$: $\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))$

Step 3: Substitute and Combine Terms

Now the expression is: $e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x))$

Using $\sec(x) = \frac{1}{\cos(x)}$, we have: $\ln(\sec(x)) = -\ln(\cos(x))$ Thus, $2 \ln(\sec(x)) = -2 \ln(\cos(x))$.

Substituting this back in, we get: $e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x)) = e^x$

Final Answer

The simplified expression is: $e^x$

Would you like further clarification on any of the steps?

Additional Questions

1. How would this simplification change if the expression had a different trigonometric function?
2. What happens if we replace $\cos^2(x)$ with $\sin^2(x)$ in the expression?
3. Can you simplify the expression if it included a different logarithmic function base, like $\log_{10}$?
4. What other trigonometric identities can help simplify similar expressions?
5. How would the approach differ if $\tan(x)$ was replaced by $\cot(x)$?

Tip

Remember that recognizing trigonometric identities, such as $1 + \tan^2(x) = \sec^2(x)$, can significantly simplify complex expressions.