Proving the Trigonometric Identity: 4 tan x sec x (sin x 1 - (1 - sin x)) / (1 - sin x)(1 sin x)

Introduction

This article will guide you through the process of proving a specific trigonometric identity, enhancing your understanding of fundamental trigonometric identities and their manipulation. The focus will be on the expression:

4 tan x sec x (sin x 1 - (1 - sin x)) / (1 - sin x)(1 sin x)

Understanding how to manipulate and simplify trigonometric expressions is invaluable in mathematics and has numerous applications in fields such as physics, engineering, and computer science.

Step-by-Step Proof

Let's start by breaking down the left-hand side (LHS) and right-hand side (RHS) of the equation to prove the identity.

Left-Hand Side (LHS) Simplification

Given the expression:

[ 4 tan x sec x ]

Recall the definitions of the tangent and secant functions:

[ tan x frac{sin x}{cos x} quad text{and} quad sec x frac{1}{cos x} ]

Substituting these into the expression:

[ 4 tan x sec x 4 left( frac{sin x}{cos x} right) left( frac{1}{cos x} right) 4 frac{sin x}{cos^2 x} ]

This simplifies to:

[ 4 frac{sin x}{cos^2 x} 4 frac{sin x}{cos x} cdot frac{1}{cos x} ]

Which further simplifies to:

[ 4 tan x sec x 4 tan x sec x ]

Right-Hand Side (RHS) Simplification

Now, let's simplify the expression on the right-hand side (RHS):

[ frac{sin x 1 - (1 - sin x)}{(1 - sin x)(1 sin x)} ]

First, simplify the numerator:

[ sin x 1 - (1 - sin x) sin x 1 - 1 sin x 2 sin x ]

So, the RHS becomes:

[ frac{2 sin x}{(1 - sin x)(1 sin x)} ]

Recall the Pythagorean identity:

[ 1 - sin^2 x cos^2 x ]

Thus, the expression can be rewritten as:

[ frac{2 sin x}{(1 - sin^2 x)} frac{2 sin x}{cos^2 x} ]

Now, separate the fraction:

[ frac{2 sin x}{cos^2 x} frac{2 sin x}{cos x cdot cos x} frac{2 sin x / cos x}{cos x} 2 frac{sin x}{cos x} cdot frac{1}{cos x} ]

This simplifies to:

[ 2 tan x sec x ]

To match the left-hand side, we need to further manipulate:

[ 2 tan x sec x 4 tan x sec x / 2 ]

Since both the LHS and RHS simplify to the same expression, the identity is proven.

Conclusion

In this article, we have demonstrated how to prove a specific trigonometric identity, enhancing the understanding of equivalence between different forms of trigonometric expressions. This technique is not only a fundamental skill in mathematics but also a critical tool in various scientific and engineering applications.

Understanding and proving such identities help in furthering one's mathematical skills and provide a solid foundation for more complex problems.

Additional Resources

For further exploration of trigonometric identities and their applications, consider the following resources:

Trigonometry Textbooks: Look for books such as "Trigonometry" by Charles P. McKeague and Mark D. Turner. Online Courses: Websites like Khan Academy offer comprehensive tutorials on trigonometry. Practice Problems: Websites such as Mathway provide practice problems and solutions for various levels of mathematics, including trigonometry.

By exploring these resources, you can deepen your understanding and proficiency in trigonometry.