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


We are given the expression:

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

Let's simplify this step by step.

### Step 1: Simplifying $ \ln(\cos^2(x)) $
We use the logarithmic property $ \ln(a^b) = b\ln(a) $, where $ a = \cos(x) $ and $ b = 2 $. Thus,

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

So the expression becomes:

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

### Step 2: Simplifying $ \ln(1 + \tan^2(x)) $
Recall the trigonometric identity $ 1 + \tan^2(x) = \sec^2(x) $. Using this identity, we have:

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

Now the expression becomes:

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

### Step 3: Using the identity $ \sec(x) = \frac{1}{\cos(x)} $
Next, we use the fact that $ \sec(x) = \frac{1}{\cos(x)} $. Therefore, we can rewrite $ \ln(\sec(x)) $ as:

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

Substituting this into the expression, we get:

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

This simplifies further to:

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

### Final Simplified Expression:
Thus, the simplified expression is:

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

---

Would you like a deeper explanation of any steps, or do you have further questions?

### Related questions:
1. How does the identity $ 1 + \tan^2(x) = \sec^2(x) $ come from the Pythagorean identity?
2. Can the logarithmic properties used here be applied to other trigonometric functions?
3. What happens if we differentiate this simplified expression with respect to $ x $?
4. How would this expression change if the original had $ \sin^2(x) $ instead of $ \cos^2(x) $?
5. Could we simplify the expression further if $ x $ had specific values like $ x = 0 $?

#### Tip:
Always remember to apply the logarithmic properties carefully, especially when dealing with powers and products of functions inside the logarithm!

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

This guide walks you through simplifying the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) step by step. Learn how to apply logarithmic properties and trigonometric identities for a complete solution. Suitable for advanced high school students in Grades 11-12.

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