e^x-ln(cos^2(x))+ln(1+tan^2(x)) simplify to single term

We are given the expression:

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

Let's simplify it step by step.

### Step 1: Simplify the logarithmic terms

First, recognize that \( \ln(1 + \tan^2(x)) \) has a well-known trigonometric identity:

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

So, we can rewrite that part of the expression:

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

Now we have:

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

### Step 2: Use the identity \( \sec(x) = \frac{1}{\cos(x)} \)

Since \( \sec^2(x) = \frac{1}{\cos^2(x)} \), we can substitute this into the expression:

\[
e^x - \ln(\cos^2(x)) + \ln\left(\frac{1}{\cos^2(x)}\right)
\]

### Step 3: Combine the logarithmic terms

Notice that \( \ln\left(\frac{1}{\cos^2(x)}\right) = -\ln(\cos^2(x)) \). So, the expression simplifies to:

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

This simplifies further:

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

### Step 4: Apply the logarithmic property

Using the property \( \ln(a^b) = b\ln(a) \), we rewrite \( 2\ln(\cos^2(x)) \) as:

\[
2 \ln(\cos(x))
\]

Thus, the expression becomes:

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

At this point, the expression is simplified to its most compact form:

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

This is the simplified single term expression.

Would you like further details or clarification on any part of this process?

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Here are 5 questions related to this topic:
1. How does the property \( \ln(a^b) = b\ln(a) \) work in logarithmic equations?
2. What is the relationship between \( \tan(x) \) and \( \sec(x) \) in trigonometry?
3. Why is \( 1 + \tan^2(x) = \sec^2(x) \) such a commonly used identity?
4. Can the expression \( e^x - 2\ln(\cos(x)) \) be further transformed or simplified?
5. How does the natural logarithm \( \ln \) differ from common logarithms?

**Tip**: Always look for common trigonometric identities when simplifying expressions with logarithms and trigonometric functions.

e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) simplify to single term

This step-by-step guide simplifies the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using key mathematical concepts such as logarithmic properties and trigonometric identities. Suitable for advanced high school students in Grades 11-12.

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