Consider the expression e^x - In(cos²(x)) + In(1 + tan^2(x)) and then simplify it into a single term

To simplify the expression 

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

we can proceed by working on the logarithmic terms and trigonometric identities. 

### Step-by-Step Simplification
1. **Simplify the Logarithmic Terms:**
   Recall the identity \(1 + \tan^2(x) = \sec^2(x)\). So, we have:
   \[
   \ln(1 + \tan^2(x)) = \ln(\sec^2(x)).
   \]

2. **Combine the Logarithms:**
   Now the expression becomes:
   \[
   e^x - \ln(\cos^2(x)) + \ln(\sec^2(x)).
   \]
   Using the fact that \(\sec(x) = \frac{1}{\cos(x)}\), we get \(\sec^2(x) = \frac{1}{\cos^2(x)}\).

3. **Rewrite \(\ln(\sec^2(x))\) in Terms of \(\cos(x)\):**
   Substitute \(\sec^2(x) = \frac{1}{\cos^2(x)}\) into the logarithm:
   \[
   \ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right) = -\ln(\cos^2(x)).
   \]

4. **Combine Terms:**
   Substitute this back into the expression:
   \[
   e^x - \ln(\cos^2(x)) - \ln(\cos^2(x)).
   \]
   Simplifying, we have:
   \[
   e^x - 2\ln(\cos(x)).
   \]

5. **Final Simplified Form:**
   Using the logarithmic identity \(a \ln(b) = \ln(b^a)\), we get:
   \[
   e^x - \ln(\cos^2(x)) = e^x - \ln(\cos^2(x)) = e^x.
   \]



Consider the expression e^x - ln(cos²(x)) + ln(1 + tan^2(x)) and then simplify it into a single term.

Learn how to simplify the expression e^x - ln(cos²(x)) + ln(1 + tan^2(x)) using trigonometric identities and logarithmic properties. This solution provides a step-by-step approach, covering key concepts in logarithmic and trigonometric simplification, suitable for students in Grades 11-12.

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