Let's simplify the given expression step by step:

The expression is:

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

Step 1: Simplify the logarithmic terms

We start by simplifying the logarithms. The second term, $\ln(\cos^2(x))$, can be rewritten using the logarithmic property:

$\ln(a^b) = b \cdot \ln(a)$

So,

$\ln(\cos^2(x)) = 2 \cdot \ln(\cos(x))$

Thus, 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:

$1 + \tan^2(x) = \sec^2(x)$

This simplifies the third term as:

$\ln(1 + \tan^2(x)) = \ln(\sec^2(x))$

Using the logarithmic rule again:

$\ln(\sec^2(x)) = 2 \cdot \ln(\sec(x))$

Now the expression becomes:

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

Step 3: Simplify using $\sec(x) = \frac{1}{\cos(x)}$

We know that:

$\sec(x) = \frac{1}{\cos(x)}$

So:

$\ln(\sec(x)) = \ln\left(\frac{1}{\cos(x)}\right) = -\ln(\cos(x))$

Thus,

$2 \ln(\sec(x)) = -2 \ln(\cos(x))$

Step 4: Final simplification

Substitute this result into the expression:

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

This simplifies to:

$e^x - 4 \ln(\cos(x))$

Thus, the simplified expression is:

$e^x - 4 \ln(\cos(x))$

Final Result:

The expression simplifies to:

$e^x - 4 \ln(\cos(x))$

Would you like more details or have any questions?

Related Questions:

1. What are other identities that involve $\tan(x)$ and $\sec(x)$?
2. How does the logarithmic property $\ln(a^b)$ apply in other contexts?
3. How would the expression change if the cosine term were sine instead?
4. Can you explore similar simplifications for hyperbolic functions?
5. How would the expression change if we included a different exponential function?

Tip:

Always remember that logarithmic properties and trigonometric identities can be powerful tools when simplifying complex expressions.