Proving the Logarithmic Identity: log10(secx tanx) -log10(secx - tanx)

In this article, we will explore the method to prove the identity between the logarithmic expressions log10(secx tanx) and -log10(secx - tanx). We will go through each step carefully, ensuring clarity and detail in the process.

Introduction to the Problem

The identity we aim to prove is:

log10(secx tanx) -log10(secx - tanx)

Step-by-Step Proof

1. Left Hand Side (LHS)

Let's start with the left hand side of the equation:

log10(secx tanx)

2. Manipulating the Expression Inside the Logarithm

To simplify the expression, we can multiply both the numerator and the denominator by the conjugate of secx - tanx, which is secx tanx. However, we will use the simpler approach as provided:

log10(secx tanx)
Multiply both numerator and denominator of the expression inside the log by secx - tanx:

log10left{frac{sec^2{x} - tan^2{x}}{sec{x} - tan{x}}right}

3. Applying the Trigonometric Identity

Next, we use the trigonometric identity for the difference of squares:

sec^2{x} - tan^2{x} 1

Substituting this identity into our expression, we get:

log10left{frac{1}{sec{x} - tan{x}}right}

4. Simplifying the Expression

Using the logarithmic property that log10(1/a) -log10(a), we can simplify the expression as follows:

log10(1/(secx - tanx)) -log10(secx - tanx)

Conclusion

Hence, we have successfully shown that:

log10(secx tanx) -log10(secx - tanx)

Understanding the Proof Step-by-Step

Let's break down each step again for clarity:

log10(secx tanx) Multiply both numerator and denominator of the expression inside the log by secx - tanx: Use the identity sec^2{x} - tan^2{x} 1 to simplify the expression inside the logarithm. Use the logarithmic property log10(1/a) -log10(a) to simplify the final result.

Related Keywords

This proof introduces several important mathematical concepts, including:

Logarithmic Identity: A fundamental property of logarithms. secx tanx: The product of secant and tangent of an angle. Proving Logarithmic Expressions: Techniques to verify and manipulate logarithmic expressions.

Understanding these concepts and techniques is crucial for anyone dealing with logarithmic functions and trigonometry.