Simplifying Trigonometric Expressions: Understanding the Value of sin2x / (1 cosx - cos2x)

When working with trigonometric equations, it's important to have a solid understanding of how to simplify and manipulate these expressions. This article will guide you through the process of evaluating the expression sin2x / (1 cosx - cos2x), explaining each step along the way and highlighting the key trigonometric identities used in the derivation.

Simplifying the Denominator

The expression we are working with is sin2x / (1 cosx - cos2x). First, let's focus on the denominator. We need to simplify 1 cos x - cos2x.

Step 1: Common Denominator and Simplify

Start by taking the common denominator, which is 1 cos x. This gives us:

1/cosx1-cosx sin2x/(1-cos2x)

Recall that 1 - cos2x sin2x. Therefore, we can rewrite the denominator as:

sin2x / sin2x

Step 2: Further Simplification

Since the numerator and the denominator are the same, this simplifies to 1. Thus, the expression reduces to:

Understanding the Process

To better understand this process, let's break it down into clear steps:

Step 1: Given Expression

We start with the expression sin2x / (1 cosx - cos2x).

Step 2: Simplify the Denominator

Recall the trigonometric identity:

1 - cos2x sin2x

We can now rewrite the denominator using this identity:

1 cosx - cos2x 1 cosx - (1 - sin2x)

This simplifies to:

1 cosx - 1 sin2x cosx sin2x

Since we need a common denominator, we can factor out a 1 - cosx:

cosx sin2x sin2x/(1 - cosx)

Step 3: Final Simplification

Now, multiply the numerator and denominator by 1 - cosx:

sin2x / (1 - cos2x)

Recall that 1 - cos2x sin2x:

sin2x / sin2x 2

Key Formulas and Identies

Throughout this process, we used the following key trigonometric identities: