e 
x
 −ln(cos 
2
 (x))+ln(1+tan 
2
 (x))

Let's simplify the expression step by step:

### Given Expression:

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

### Step 1: Simplify \(\ln(\cos^2(x))\)
We know the property of logarithms:

\[
\ln(a^b) = b \ln(a)
\]

Applying this to \(\ln(\cos^2(x))\):

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

### Step 2: Simplify \(\ln(1 + \tan^2(x))\)
Using the trigonometric identity:

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

Thus:

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

Again, using the property of logarithms:

\[
\ln(\sec^2(x)) = 2\ln(\sec(x))
\]

### Step 3: Combine the expression

Now, substitute these simplified forms back into the original expression:

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

### Step 4: Simplify \(\ln(\sec(x))\)
Recall that:

\[
\sec(x) = \frac{1}{\cos(x)}
\]

So:

\[
\ln(\sec(x)) = \ln\left(\frac{1}{\cos(x)}\right) = -\ln(\cos(x))
\]

### Step 5: Final Expression

Substituting \(\ln(\sec(x)) = -\ln(\cos(x))\) back into the expression:

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

Simplify further:

\[
e^x - 2\ln(\cos(x)) - 2\ln(\cos(x)) = e^x - 4\ln(\cos(x))
\]

### Final Simplified Form:

\[
e^x - 4\ln(\cos(x))
\]

This is the fully simplified form of the given expression.

---

Would you like further explanation, or do you have any questions?

Here are some related questions you might want to explore:

1. What is the derivative of this expression?
2. Can you integrate this expression with respect to \(x\)?
3. How would this expression change if we used different trigonometric identities?
4. Can this expression be further simplified under specific conditions for \(x\)?
5. What happens when \(x = 0\) in the simplified expression?

**Tip**: When simplifying logarithmic expressions, always look for opportunities to apply logarithmic properties and trigonometric identities!

Discover how to simplify the expression e^x − ln(cos^2(x)) + ln(1 + tan^2(x)) step by step using logarithmic properties and trigonometric identities. This guide explores exponential functions, trigonometric simplifications, and logarithmic rules, perfect for advanced high school or undergraduate math students.

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