The image you've uploaded includes a mathematical expression and its simplification. Let's break down the steps mentioned:

The original expression is: $e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))$

Step-by-step explanation:

1. Simplification of the second term: $- \ln(\cos^2(x))$ Using the logarithmic property $\ln(a^b) = b \ln(a)$, we rewrite it as: $-2 \ln(\cos(x))$

2. Simplification of the third term: $\ln(1 + \tan^2(x))$ Recall the trigonometric identity: $1 + \tan^2(x) = \sec^2(x)$ So, this term becomes: $\ln(\sec^2(x)) = 2 \ln(\sec(x))$

3. Putting it all together: Now the expression becomes: $e^x - 2 \ln(\cos(x)) + 2 \ln(\sec(x))$

4. Simplifying further: Use the identity $\sec(x) = \frac{1}{\cos(x)}$, so: $2 \ln(\sec(x)) = 2 \ln\left(\frac{1}{\cos(x)}\right) = -2 \ln(\cos(x))$

5. Final result: Now the expression simplifies to: $e^x$ Since the logarithmic terms cancel each other out.

Would you like more details on any of these steps, or have any further questions?

Related questions:

1. How do logarithmic properties work in general expressions?
2. Can you explain the trigonometric identities used in this problem?
3. What other simplifications can be done using logarithmic identities?
4. How does the identity $\sec(x) = \frac{1}{\cos(x)}$ affect logarithmic expressions?
5. Can you show examples of similar expressions involving both $e^x$ and logarithms?

Tip:

Always watch for trigonometric identities when simplifying expressions involving logarithms of trigonometric functions—they can significantly simplify your work!