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:
2
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:
1 - cos2x sin2xThese identities are fundamental in simplifying and solving more complex trigonometric equations.
Conclusion
Simplifying trigonometric expressions like sin2x / (1 cosx - cos2x) can be achieved by carefully using basic trigonometric identities. This step-by-step approach ensures that each part of the expression is simplified effectively, leading to the final answer of 2. Understanding these techniques is crucial for anyone working with trigonometric equations.