Deriving the Value of sec A tan A -1 / (tan A - sec A): A Comprehensive Guide

In this article, we discuss the derivation of the trigonometric expression sec A tan A -1 / (tan A - sec A). Understanding the transformation and simplification of such expressions is crucial for students and professionals in fields such as engineering, physics, and mathematics. We will explore the step-by-step simplification process, along with the necessary trigonometric identities and concepts.

Introduction to Trigonometric Functions

Before we dive into the derivation, it is essential to understand the basic definitions of the trigonometric functions involved:

sec A: The secant of an angle A is defined as 1 / cos A. tan A: The tangent of an angle A is defined as sin A / cos A.

The Step-by-Step Derivation

We start with the expression:

sec A tan A -1 / (tan A - sec A)

We can rewrite the numerator and the denominator using the definitions of sec A and tan A:

sec A 1 / cos A

tan A sin A / cos A

Substituting these values, we get:

sec A tan A - 1 / (tan A - sec A) (1 / cos A) * (sin A / cos A) - 1 / (sin A / cos A - 1 / cos A)

Further simplifying, we multiply the numerator and denominator by cos A^2:

sec A tan A -1 / (tan A - sec A) (1 * sin A - cos A) / (sin A - cos A)

We now have:

(1 * sin A - cos A) / (sin A - cos A)

To simplify further, we can use the double-angle identities for sine and cosine:

cos 2A 1 - 2sin^2 A

sin 2A 2sin A cos A

We redefine our angle from A to A/2:

sec A tan A -1 / (tan A - sec A) (1 - 2sin^2(A/2) - cos(2A/2)) / (2sin(A/2)cos(A/2) - 1 - cos(2A/2))

The 1 cancels out, and we are left with:

(2sin(A/2)cos(A/2) - sin^2(A/2)) / (2sin(A/2)cos(A/2) - 1 - 2sin^2(A/2))

We then divide the numerator and denominator by 2sin(A/2)cos(A/2):

(cos(A/2) - sin(A/2)) / 1 - tan(A/2)

This simplifies to:

tan(A/2) / (1 - tan(A/2))

Thus, we have derived the value of:

sec A tan A -1 / (tan A - sec A) tan(A/2) / (1 - tan(A/2))

Conclusion

In conclusion, the expression sec A tan A -1 / (tan A - sec A) simplifies to tan(A/2) / (1 - tan(A/2)). This derivation involves the use of basic trigonometric identities and can be an essential tool for solving more complex trigonometric equations. Understanding such expressions and their simplifications is crucial for many practical applications in mathematics, physics, and engineering. We encourage readers to practice similar problems to deepen their understanding of these concepts.

References

Math is Fun - Trigonometry Math is Fun - Trigonometric Identities