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


To simplify the expression \( e - \ln(\cos^2(x)) + \ln(1 + \tan(x)) \), let's proceed step by step.

1. **Rewrite the logarithmic terms**:
   \[
   -\ln(\cos^2(x)) + \ln(1 + \tan(x))
   \]
   Using the properties of logarithms, we can combine these:
   \[
   = \ln\left(\frac{1 + \tan(x)}{\cos^2(x)}\right)
   \]

2. **Simplify the expression inside the logarithm**:
   Recall that \( 1 + \tan(x) = \frac{\sin(x) + \cos(x)}{\cos(x)} \) and \( \cos^2(x) = \cos(x) \cdot \cos(x) \).
   Thus, we get:
   \[
   \frac{1 + \tan(x)}{\cos^2(x)} = \frac{\frac{\sin(x) + \cos(x)}{\cos(x)}}{\cos^2(x)} = \frac{\sin(x) + \cos(x)}{\cos

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

Explore how to simplify the expression e - ln(cos²(x)) + ln(1 + tan(x)) into a single term. This solution covers logarithmic properties, trigonometric identities, and step-by-step transformations suitable for advanced math students in Grades 11-12.

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