Consider the expression e^x-ln(Cos^2(x))+ln(1+tan^2(x))

The given expression is:

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

Let's simplify it step by step.

1. **Simplifying the logarithmic terms:**

   - \(\ln(\cos^2(x))\) can be rewritten using the logarithmic property \(\ln(a^b) = b\ln(a)\):
     \[
     \ln(\cos^2(x)) = 2\ln(\cos(x))
     \]

   - Now consider \(\ln(1 + \tan^2(x))\). Using the identity \(1 + \tan^2(x) = \sec^2(x)\), we get:
     \[
     \ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2\ln(\sec(x))
     \]

2. **Substituting into the original expression:**
   \[
   e^x - 2\ln(\cos(x)) + 2\ln(\sec(x))
   \]

3. **Simplifying further:**

   - Since \(\sec(x) = \frac{1}{\cos(x)}\), \(\ln(\sec(x)) = -\ln(\cos(x))\).
   - Therefore, \(2\ln(\sec(x)) = -2\ln(\cos(x))\).

4. **Now the expression becomes:**
   \[
   e^x - 2\ln(\cos(x)) + (-2\ln(\cos(x)))
   \]
   This simplifies to:
   \[
   e^x
   \]

Thus, the simplified expression is just:

\[
e^x
\]

Let me know if you need more details or further clarification!

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Here are 5 related questions to expand your understanding:
1. What is the derivative of \( e^x \)?
2. How does the identity \( 1 + \tan^2(x) = \sec^2(x) \) arise from trigonometric relationships?
3. How can logarithmic properties help simplify complex expressions?
4. What is the integral of \( e^x \)?
5. How would the simplification change if \(\tan^2(x)\) were replaced by another trigonometric function?

**Tip:** Always look for trigonometric identities and logarithmic properties to simplify complex expressions!

Consider the expression e^x - ln(Cos^2(x)) + ln(1 + tan^2(x))

This page provides a step-by-step simplification of the expression e^x - ln(Cos^2(x)) + ln(1 + tan^2(x)). Key concepts include logarithmic properties and trigonometric identities such as 1 + tan^2(x) = sec^2(x). Suitable for students in Grades 11-12 studying advanced algebra and trigonometry.

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