The expression shown in the image is:

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

Let's simplify this step by step:

Step 1: Use a trigonometric identity

We can use the identity:

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

Thus, the expression becomes:

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

Step 2: Simplify the logarithms

Recall that $\ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right)$. So the expression becomes:

$e^x - \ln(\cos^2(x)) + \ln\left(\frac{1}{\cos^2(x)}\right)$

Step 3: Combine the logarithms

Now we can combine the logarithms:

$e^x - \ln(\cos^2(x)) + \ln\left(\frac{1}{\cos^2(x)}\right) = e^x - \ln(\cos^2(x)) + (-\ln(\cos^2(x))) = e^x$

Final Simplified Expression:

The expression simplifies to just:

$e^x$

Would you like further explanation on any step or more details?

Here are 5 related questions to deepen your understanding:

1. How does the identity $1 + \tan^2(x) = \sec^2(x)$ simplify trigonometric expressions?
2. Why does $\ln(\sec^2(x))$ become $\ln\left(\frac{1}{\cos^2(x)}\right)$?
3. Can you explain the properties of logarithms used to combine the terms?
4. What happens if a similar expression involved a sine or tangent function?
5. How does the simplification process change for more complex trigonometric-logarithmic combinations?

Tip: Always look for opportunities to apply trigonometric identities in combination with logarithmic properties to simplify expressions efficiently!