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

1. **Using the trigonometric identity for tangent**:
   \[
   1 + \tan^2(x) = \sec^2(x)
   \]
   So the expression \( \ln(1 + \tan^2(x)) \) becomes:
   \[
   \ln(\sec^2(x))
   \]
   Since \( \ln(a^b) = b \ln(a) \), we can rewrite this as:
   \[
   \ln(\sec^2(x)) = 2\ln(\sec(x))
   \]

2. **Simplifying \( \ln(\cos^2(x)) \)**:
   Similarly, apply the logarithmic property \( \ln(a^b) = b\ln(a) \):
   \[
   \ln(\cos^2(x)) = 2\ln(\cos(x))
   \]

3. **Substitute back into the original expression**:
   The expression now becomes:
   \[
   e^x - 2\ln(\cos(x)) + 2\ln(\sec(x))
   \]

4. **Simplifying \( \ln(\sec(x)) \)**:
   Recall that \( \sec(x) = \frac{1}{\cos(x)} \), so:
   \[
   \ln(\sec(x)) = \ln\left(\frac{1}{\cos(x)}\right) = -\ln(\cos(x))
   \]
   Therefore:
   \[
   2\ln(\sec(x)) = -2\ln(\cos(x))
   \]

5. **Final simplification**:
   Now substitute \( 2\ln(\sec(x)) = -2\ln(\cos(x)) \) into the expression:
   \[
   e^x - 2\ln(\cos(x)) + (-2\ln(\cos(x))) = e^x
   \]

So, the simplified expression is:

\[
e^x
\]

Would you like further details or clarifications?

---

Here are 5 questions related to this simplification:

1. How can you apply logarithmic properties in other complex expressions?
2. What other trigonometric identities can be useful in simplifying logarithmic expressions?
3. Why does the logarithm of a reciprocal lead to a negative value?
4. What is the significance of the trigonometric identity \( 1 + \tan^2(x) = \sec^2(x) \)?
5. How does this simplification work in real-life applications involving exponential and logarithmic functions?

**Tip**: Always check for trigonometric identities when simplifying expressions involving trigonometric functions. They can significantly simplify the process!

Learn how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic properties and trigonometric identities. This step-by-step guide covers key exponential, logarithmic, and trigonometric concepts, suitable for advanced high school or college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation