Let's simplify the expression $e - \ln(\cos(x)) + \ln(1 + \tan^2(x))$ step by step.

1. Break the expression into individual parts:

• $e$ is just a constant.
• $\ln(\cos(x))$ is the natural logarithm of $\cos(x)$.
• $\ln(1 + \tan^2(x))$ is the natural logarithm of $1 + \tan^2(x)$.

2. Use a known trigonometric identity:

There is a well-known identity: $1 + \tan^2(x) = \sec^2(x)$ Thus, we can replace $1 + \tan^2(x)$ with $\sec^2(x)$: $\ln(1 + \tan^2(x)) = \ln(\sec^2(x))$

3. Simplify $\ln(\sec^2(x))$:

Recall that $\sec(x) = \frac{1}{\cos(x)}$, so: $\ln(\sec^2(x)) = \ln\left(\frac{1}{\cos^2(x)}\right)$ Using the property of logarithms $\ln(a^b) = b \ln(a)$, this becomes: $\ln(\sec^2(x)) = 2 \ln\left(\frac{1}{\cos(x)}\right) = 2(-\ln(\cos(x))) = -2\ln(\cos(x))$

4. Substitute back into the original expression:

Now, substitute this back into the expression: $e - \ln(\cos(x)) + \ln(1 + \tan^2(x)) = e - \ln(\cos(x)) - 2\ln(\cos(x))$ Simplify: $e - 3\ln(\cos(x))$

Final Answer:

The simplified expression is: $e - 3\ln(\cos(x))$

Would you like more details or clarifications on any step?

Here are 5 related questions:

1. What are the basic properties of logarithms?
2. How does the trigonometric identity $1 + \tan^2(x) = \sec^2(x)$ arise from the Pythagorean identity?
3. How do you differentiate $\ln(\cos(x))$?
4. What is the Taylor expansion of $\ln(\cos(x))$ around $x = 0$?
5. How does the constant $e$ interact with logarithmic functions in calculus?

Tip: Always check for trigonometric identities when simplifying logarithmic expressions involving trigonometric functions.