Simplify e^x-ln(cos^2(x))+ln(1+tan^2(x)) as a single expression using trigonometric substitution.

To simplify the given expression \( e^x - \ln(\cos^2(x)) + \ln(1 + \tan^2(x)) \), let's work through the steps methodically and apply appropriate trigonometric identities.

### Step 1: Use the identity for \(1 + \tan^2(x)\)
We know from trigonometry that:
\[
1 + \tan^2(x) = \sec^2(x)
\]
Thus, the term \( \ln(1 + \tan^2(x)) \) becomes:
\[
\ln(1 + \tan^2(x)) = \ln(\sec^2(x))
\]

### Step 2: Simplify the logarithmic terms
We now rewrite the two logarithmic terms:

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

Recall that \( \sec(x) = \frac{1}{\cos(x)} \). Thus:
\[
\ln(\sec(x)) = -\ln(\cos(x))
\]
So:
\[
\ln(\sec^2(x)) = 2(-\ln(\cos(x))) = -2\ln(\cos(x))
\]

### Step 3: Combine the logarithmic terms
Now substitute back into the original expression:
\[
e^x - 2\ln(\cos(x)) + 2(-\ln(\cos(x)))
\]
This simplifies further:
\[
e^x - 2\ln(\cos(x)) - 2\ln(\cos(x)) = e^x
\]

### Final Simplified Expression:
Thus, the expression simplifies to:
\[
e^x
\]

### Conclusion:
The given expression simplifies to just \( e^x \).

---
Would you like more details or have any questions about this simplification?

Here are 5 related questions you might find helpful:
1. How is the logarithmic property \( \ln(a^b) = b\ln(a) \) used in simplification?
2. What is the importance of trigonometric identities in simplifying expressions?
3. Why does \( \ln(\sec^2(x)) \) simplify to \( -2\ln(\cos(x)) \)?
4. Can similar methods be applied to simplify expressions involving \( \sin(x) \) and \( \cot(x) \)?
5. What is the derivative of \( e^x \) and how is it applied in calculus problems?

**Tip:** When simplifying logarithmic expressions, always check if you can use properties like \( \ln(a \cdot b) = \ln(a) + \ln(b) \) to break down complex terms.

Simplify e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) as a single expression using trigonometric substitution.

Simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using trigonometric identities. This step-by-step solution utilizes key concepts such as logarithmic properties and trigonometric simplification, resulting in the expression e^x. Suitable for precalculus and calculus students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation