Problem 95 - Precalculus Essentials

Question

Verify each identity. $$\ln e^{\tan ^{2} x-\sec ^{2} x}=-1$$

Step-by-Step Solution

Answer

Yes, the identity \( \ln e^{ tan ^{2} x-\sec ^{2} x}=-1 \) is verified.

Step 1: Understanding the Problem

The initial step is understanding an identity. An identity in mathematics is an equation that remains true for all values of the variables. We have to verify the identity \( \ln e^{\tan ^{2} x-\sec ^{2} x}=-1 \). This means we have to demonstrate that the left-hand side expression is equal to the right-hand side expression, which is -1 in this case.

Step 2: Using Properties of Logarithms and Trigonometric Functions

Now consider the left-hand side, \( \ln e^{\tan ^{2} x-\sec ^{2} x} \). According to property of logarithms, if \( \ln a^{n} \), where \( a > 0 \) and \( n > 0 \), equals \( n \cdot \ln a \). Now apply this property to \( \ln e^{\tan ^{2} x-\sec ^{2} x} \) to get \( (\tan ^{2} x-\sec ^{2} x) \cdot \ln e \). Also, note that \( \ln e = 1 \), so the left-hand side becomes \( \tan ^{2} x-\sec ^{2} x \).

Step 3: Further Simplification using Pythagorean Identity

Next, remember that \( \tan ^{2} x+1=\sec ^{2} x \). Therefore, substituting this identity into the simplifed left-hand side expression gives \( \tan ^{2} x - (\tan ^{2} x + 1) \) which simplifies to \( -1 \).