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

We are tasked with simplifying the expression:

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

### Step 1: Use the identity for $\tan^2(x)$

Recall that:

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

Thus, the expression becomes:

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

### Step 2: Simplify the logarithmic terms

Using the property of logarithms $\ln(a^b) = b\ln(a)$, we simplify both $\ln(\cos^2(x))$ and $\ln(\sec^2(x))$:

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

### Step 3: Simplify further

Since $\sec(x) = \frac{1}{\cos(x)}$, we can use the logarithm rule $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$ to simplify $\ln(\sec(x)) = -\ln(\cos(x))$. Applying this:

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

Thus, the expression simplifies completely to:

\[
e^x
\]

### Final Answer:
The simplified form of the expression is:

\[
e^x
\]

Would you like further details or have any questions?

Here are 5 related questions you can explore:
1. How would the expression change if we replaced $\tan^2(x)$ with $\cot^2(x)$?
2. What properties of logarithms are most useful for simplifying expressions?
3. Can this simplification process be applied to more complex trigonometric functions?
4. How would this expression behave for specific values of $x$?
5. What other identities for $\tan(x)$ could be used in such problems?

**Tip:** Always keep fundamental trigonometric identities in mind—they often lead to significant simplifications in complex expressions!

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

Simplify the expression e^x - ln(cos^2(x)) + ln(1 + tan^2(x)) into a single term using properties of logarithms and trigonometric identities. This detailed solution is ideal for students in Grades 10-12, covering key concepts in logarithms and trigonometry.

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