Consider the expression e ^ x - ln(cos^2 (x)) + ln(1 + tan^2 (x)) and then simplify it into a single term

We are given the expression:

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

We will simplify it step by step.

### Step 1: Simplify the logarithmic terms
Use the logarithmic identity for powers:

\[
\ln(\cos^2(x)) = 2 \ln(\cos(x))
\]

So the expression becomes:

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

### Step 2: Use the identity for \( \tan^2(x) \)
Recall that:

\[
1 + \tan^2(x) = \sec^2(x)
\]

Thus, the expression now is:

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

### Step 3: Simplify \( \ln(\sec^2(x)) \)
Since \( \sec(x) = \frac{1}{\cos(x)} \), we have:

\[
\ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right) = - \ln(\cos^2(x))
\]

Using the previous result \( \ln(\cos^2(x)) = 2 \ln(\cos(x)) \), we get:

\[
\ln(\sec^2(x)) = - 2 \ln(\cos(x))
\]

### Step 4: Combine the terms
Now substitute this back into the expression:

\[
e^x - 2 \ln(\cos(x)) - 2 \ln(\cos(x))
\]

Simplify:

\[
e^x - 4 \ln(\cos(x))
\]

This is the simplified expression. There is no further simplification possible without making assumptions about the value of \( x \).

### Final simplified expression:
\[
e^x - 4 \ln(\cos(x))
\]

Would you like more details or clarifications on any step?

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Here are some related questions you can explore:
1. What are the properties of the exponential function \( e^x \)?
2. How do logarithmic identities help in simplifying expressions?
3. What is the relationship between \( \sec(x) \) and \( \cos(x) \)?
4. How can we apply this expression for specific values of \( x \)?
5. Can this expression be simplified further for small values of \( x \)?

**Tip**: When simplifying logarithms, always look for power or quotient rules to break down complicated terms easily!

Consider the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) and then simplify it into a single term

Learn how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic and trigonometric identities. This step-by-step solution demonstrates exponential functions and trigonometric identities, making it ideal for advanced high school students in Grades 11-12.

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