Consider the expression e* - In(cos (x)) + In(1 + tan? (x)) and then simplify it into a single term

Let's simplify the expression \( e - \ln(\cos(x)) + \ln(1 + \tan^2(x)) \) step by step.

### 1. Break the expression into individual parts:
- \( e \) is just a constant.
- \( \ln(\cos(x)) \) is the natural logarithm of \( \cos(x) \).
- \( \ln(1 + \tan^2(x)) \) is the natural logarithm of \( 1 + \tan^2(x) \).

### 2. Use a known trigonometric identity:
There is a well-known identity:
\[
1 + \tan^2(x) = \sec^2(x)
\]
Thus, we can replace \( 1 + \tan^2(x) \) with \( \sec^2(x) \):
\[
\ln(1 + \tan^2(x)) = \ln(\sec^2(x))
\]

### 3. Simplify \( \ln(\sec^2(x)) \):
Recall that \( \sec(x) = \frac{1}{\cos(x)} \), so:
\[
\ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right)
\]
Using the property of logarithms \( \ln(a^b) = b \ln(a) \), this becomes:
\[
\ln(\sec^2(x)) = 2 \ln\left(\frac{1}{\cos(x)}\right) = 2(-\ln(\cos(x))) = -2\ln(\cos(x))
\]

### 4. Substitute back into the original expression:
Now, substitute this back into the expression:
\[
e - \ln(\cos(x)) + \ln(1 + \tan^2(x)) = e - \ln(\cos(x)) - 2\ln(\cos(x))
\]
Simplify:
\[
e - 3\ln(\cos(x))
\]

### Final Answer:
The simplified expression is:
\[
e - 3\ln(\cos(x))
\]

Would you like more details or clarifications on any step?

Here are 5 related questions:
1. What are the basic properties of logarithms?
2. How does the trigonometric identity \(1 + \tan^2(x) = \sec^2(x)\) arise from the Pythagorean identity?
3. How do you differentiate \( \ln(\cos(x)) \)?
4. What is the Taylor expansion of \( \ln(\cos(x)) \) around \( x = 0 \)?
5. How does the constant \( e \) interact with logarithmic functions in calculus?

**Tip:** Always check for trigonometric identities when simplifying logarithmic expressions involving trigonometric functions.

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

Explore the step-by-step simplification of the expression e - ln(cos(x)) + ln(1 + tan^2(x)). This problem covers logarithmic properties, trigonometric identities, and calculus fundamentals, making it suitable for students in Grades 10-12.

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