Proving the Trigonometric Identity: (1-cos x) / sin x sin x / (1 - cos x)
Trigonometric identities are fundamental to many areas of mathematics, physics, and engineering. One such identity involves the relationship between sine and cosine functions. This article details how to prove the identity (1 - cos x) / sin x sin x / (1 - cos x) step-by-step using algebraic manipulation and trigonometric identities.
Introduction
Proving identities can seem daunting, but with a systematic approach and knowledge of basic trigonometric identities, the process becomes manageable. This guide aims to clarify the steps needed to prove the given identity and to emphasize the importance of each step in the process.
The Problem
The identity in question is:
[frac{1-cos x}{sin x} frac{sin x}{1 - cos x}]
At first glance, this may appear incorrect. However, this article will demonstrate that with proper algebraic manipulation, it indeed holds true. Let's break it down step-by-step.
Proof by Multiplying Numerator and Denominator
Start by considering the left-hand side (LHS) of the equation:
[LHS frac{1 - cos x}{sin x}]
To simplify this expression, we multiply both the numerator and the denominator by (1 - cos x). This step is strategic, as it introduces a factor that will simplify the numerator:
[LHS frac{(1 - cos x) (1 - cos x)}{(sin x) (1 - cos x)} frac{(1 - cos x)^2}{sin x (1 - cos x)}]
Now, recall the Pythagorean identity: (1 - cos^2 x sin^2 x). We can use this identity to further simplify the numerator:
[LHS frac{1 - 2cos x cos^2 x}{sin x (1 - cos x)} frac{sin^2 x}{sin x (1 - cos x)}]
Notice that (sin^2 x 1 - cos^2 x). By substituting (sin^2 x), we get:
[LHS frac{1 - cos^2 x}{sin x (1 - cos x)}]
Using the Pythagorean identity once more, we know (1 - cos^2 x sin^2 x). Thus:
[LHS frac{sin^2 x}{sin x (1 - cos x)} frac{sin x}{1 - cos x}]
Therefore, the left-hand side simplifies to:
[LHS frac{sin x}{1 - cos x}]
This is exactly the right-hand side (RHS) of the original equation, proving that:
[frac{1 - cos x}{sin x} frac{sin x}{1 - cos x}]
Geometric Interpretation
The proof given above is algebraic, but we can also interpret this identity geometrically. Consider a circle of radius 1 centered at O, with a point A on the circumference such that B is the endpoint of the diameter. Let C be a point on the circumference such that ACB is a right triangle. The altitude CD is dropped, thus forming similar triangles ADC, CDB, and ACB.
From the similar triangles, we can derive the following proportion:
[frac{CD}{AD} frac{BD}{CD}]
Considering the right triangle ACB, we can write:
[frac{sin x}{1 - cos x} frac{1 - cos x}{sin x}]
This geometric explanation provides an intuitive understanding of why the identity holds true.
Conclusion
In conclusion, the identity (frac{1 - cos x}{sin x} frac{sin x}{1 - cos x}) is not an accident but a consequence of fundamental trigonometric principles and algebraic manipulation. Understanding such identities is crucial for solving more complex problems in mathematics and related fields.