Power[e,x ]-ln (Power[cos,2](x)) + ln ( 1+ Power[tan,2](x))

The given expression is:

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

Let's simplify it step by step:

### Step 1: Simplifying the logarithmic terms

1. **The second term**:
   \[
   \ln(\cos^2(x)) = 2 \ln(\cos(x))
   \]
   Using the logarithmic property \(\ln(a^b) = b \ln(a)\).

2. **The third term**:
   \[
   \ln(1 + \tan^2(x))
   \]
   Here, we use the identity \(1 + \tan^2(x) = \sec^2(x)\). Therefore:
   \[
   \ln(1 + \tan^2(x)) = \ln(\sec^2(x)) = 2 \ln(\sec(x))
   \]

### Step 2: Substitute and simplify

Now, substituting these into the original expression:

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

Since \(\ln(\sec(x)) = -\ln(\cos(x))\), we can replace \(2 \ln(\sec(x))\) with \(-2 \ln(\cos(x))\):

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

Thus, the expression simplifies further:

\[
e^x
\]

### Final Answer:
The entire expression simplifies to \(e^x\).

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Do you need further details or have any questions?

Here are 5 related questions you might find interesting:

1. How can we derive other trigonometric identities involving logarithms?
2. Can we generalize this result to other exponential and logarithmic combinations?
3. What is the derivative of \(e^x\)?
4. How would the result change if \(\cos^2(x)\) were replaced by \(\sin^2(x)\)?
5. What are the properties of the natural logarithm that help simplify such expressions?

**Tip**: When simplifying logarithmic expressions, always check for possible trigonometric identities to make simplifications easier.

Power[e,x] - ln(Power[cos,2](x)) + ln(1 + Power[tan,2](x))

Learn how to simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) using logarithmic properties and trigonometric identities. This detailed solution breaks down the steps for high school calculus students.

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