Consider the expression ex - In(cos²(x)) + ln(1+tan²(x)) and then simplify it into a single term

We are given the expression:

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

To simplify this expression, let's follow the steps:

### Step 1: Combine the logarithmic terms.
We know that \(\ln(a) - \ln(b)\) can be rewritten as \(\ln\left(\frac{a}{b}\right)\). Thus, we combine the two logarithmic terms first:

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

### Step 2: Simplify using a trigonometric identity.
We recognize that \(1 + \tan^2(x)\) is a well-known trigonometric identity:

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

Thus, the logarithmic expression becomes:

\[
\ln\left(\frac{\sec^2(x)}{\cos^2(x)}\right)
\]

Since \(\sec(x) = \frac{1}{\cos(x)}\), we know that \(\sec^2(x) = \frac{1}{\cos^2(x)}\). Therefore, we have:

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

Since \(\ln(1) = 0\), the entire logarithmic part simplifies to 0.

### Step 3: Simplify the entire expression.
Now that the logarithmic part is 0, the entire expression simplifies to:

\[
e^x + 0 = e^x
\]

Thus, the simplified expression is:

\[
e^x
\]

Would you like more details or have any questions?

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Here are 5 related questions to explore further:

1. How can we simplify expressions involving other logarithmic identities?
2. What is the derivative of \( e^x \)?
3. How do the trigonometric identities relate to logarithmic expressions?
4. Can you simplify expressions that involve both exponential and trigonometric functions?
5. How do you apply logarithmic properties in solving equations?

**Tip:** When simplifying expressions with logarithms, always check for trigonometric identities to make the process easier.

Consider the expression e^x - ln(cos²(x)) + ln(1+tan²(x)) and then simplify it into a single term.

Discover how to simplify the expression e^x - ln(cos²(x)) + ln(1+tan²(x)) into a single term using logarithmic properties and trigonometric identities. This solution covers key concepts such as simplification techniques, exponential functions, and trigonometric transformations, suitable for students in Grades 11-12.

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