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:

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

Let's break it down and simplify it step by step.

### Step 1: Simplifying \(\ln(1 + \tan^2(x))\)

Recall that \(1 + \tan^2(x) = \sec^2(x)\) from the Pythagorean identity. So we have:

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

Since \(\ln(a^b) = b\ln(a)\), this becomes:

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

### Step 2: Simplifying \(- \ln(\cos^2(x))\)

Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we can write:

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

### Step 3: Combine the logarithmic terms

Now, let's combine the two logarithmic terms:

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

Since \(\sec(x) = \frac{1}{\cos(x)}\), we know that \(\ln(\sec(x)) = -\ln(\cos(x))\). Thus:

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

So the entire logarithmic part becomes:

\[
- 2\ln(\cos(x)) + (-2\ln(\cos(x))) = 0
\]

### Step 4: Final expression

After simplifying the logarithmic terms to 0, the remaining expression is just:

\[
e^x
\]

### Final Simplified Expression:

\[
e^x
\]

This is the fully simplified version of the given expression.

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Would you like more details or explanations about this simplification?

Here are five related questions you might find useful:
1. How can logarithmic properties be used to simplify more complex expressions?
2. What are the Pythagorean identities, and how are they used in trigonometry?
3. How does the logarithmic identity \(\ln(a^b) = b\ln(a)\) apply to general logarithmic functions?
4. What are some common simplifications involving \(\tan(x)\) and \(\sec(x)\)?
5. How can exponential functions like \(e^x\) be manipulated in calculus?

**Tip:** Always check for trigonometric identities like the Pythagorean identity, as they often simplify complex trigonometric expressions easily.

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

Discover how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) step-by-step. This problem utilizes trigonometric identities and logarithmic properties, making it suitable for high school students in Grades 10-12.

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