To simplify the expression $e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x))$, let’s break it down step-by-step:

Step 1: Simplify the Logarithmic Terms

1. Rewrite $\ln(\cos^2(x))$: $\ln(\cos^2(x)) = 2 \ln(\cos(x))$

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

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

   Recall that $1 + \tan^2(x) = \sec^2(x)$, so: $\ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))$

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

Step 2: Substitute $\ln(\sec(x)) = -\ln(\cos(x))$

Since $\sec(x) = \frac{1}{\cos(x)}$, we have: $\ln(\sec(x)) = -\ln(\cos(x))$

Thus, $2 \ln(\sec(x)) = -2 \ln(\cos(x))$, making the expression: $e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x))$

Step 3: Combine Terms

Combine $-2 \ln(\cos(x)) - 2 \ln(\cos(x)) = 0$, giving us: $e^x$

Final Answer

$e^x$

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Would you like a more detailed breakdown or have any questions?

Related Questions

1. How would this simplification change if $e^x$ were a different exponential function?
2. How does this process differ if the base of the logarithm changes?
3. What are common simplifications involving $\tan(x)$ and $\sec(x)$?
4. What is the effect of adding a constant term in logarithmic simplifications?
5. How would this simplification change in polar coordinates?

Tip

When simplifying logarithmic expressions, look for identities like $1 + \tan^2(x) = \sec^2(x)$ to reduce complexity.