Simplifying Trigonometric Identities: tan(θ/2) cot(θ/2)

Understanding and simplifying trigonometric identities, such as tan(θ/2) cot(θ/2), is fundamental to advanced mathematics. This article will guide you through a detailed process of simplifying this identity using basic trigonometric functions, identities, and double-angle formulas.

Introduction to Trigonometric Functions

Before diving into the identity, it is essential to recall the basic definitions of tangent and cotangent functions:

Combining the Terms

The primary goal is to simplify the expression tan(θ/2) cot(θ/2). We start by substituting the definitions of tangent and cotangent:

tan(θ/2) cot(θ/2) (sin(θ/2) / cos(θ/2)) * (cos(θ/2) / sin(θ/2))

Multiplying the numerators and denominators:

(sin^2(θ/2) cos^2(θ/2)) / (sin(θ/2) cos(θ/2))

Using the Pythagorean Identity

The next step involves using the Pythagorean identity, which states that for any angle x, sin^2 x cos^2 x 1. However, we can manipulate this identity to fit our current expression:

1 / (sin(θ/2) cos(θ/2))

Applying the Double-Angle Formula

To further simplify this expression, we use the double-angle formula for sine, which is sin θ 2 sin(θ/2) cos(θ/2). By substituting θ with 2(θ/2), we get:

2 sin(θ/2) cos(θ/2) sin θ

Rearranging this formula to isolate sin(θ/2) cos(θ/2) and substituting it back into our expression:

1 / (1/2 * sin θ)

Since 1/2 is a constant, we can simplify this to:

2 / sin θ

Final Expression

Therefore, we have successfully simplified the expression tan(θ/2) cot(θ/2) to:

tan(θ/2) cot(θ/2) 2 csc(θ)

In this article, we have explored the process of simplifying trigonometric identities using basic definitions, identities, and double-angle formulas. Understanding these steps and techniques will help in solving more complex trigonometric problems.