Simplifying Trigonometric Expressions: A Comprehensive Guide to Simplifying sin2x-cos2x-1 / (sin2x* cos2x - 1)
Introduction to Trigonometric Identities and Simplification
Trigonometric identities are fundamental in simplifying and solving complex trigonometric expressions. In this article, we will explore how to simplify the expression (frac{sin 2x - cos 2x - 1}{sin 2x cdot cos 2x - 1}) using trigonometric identities. This process involves breaking down the expression into simpler parts and applying well-known identities to arrive at a more manageable form.Available Identities
To simplify the given expression, we can use the following trigonometric identities: (sin 2x 2sin x cos x) (cos 2x 2cos^2 x - 1 1 - 2sin^2 x)Step-by-Step Simplification
Let's start by rewriting the numerator and the denominator using these identities.Numerator: (sin 2x - cos 2x - 1)
Substitute (sin 2x) and (cos 2x): [sin 2x - cos 2x - 1 2 sin x cos x - (2 cos^2 x - 1) - 1] Simplify the expression: [sin 2x - cos 2x - 1 2 sin x cos x - 2 cos^2 x 1 - 1 2 sin x cos x - 2 cos^2 x]Substitute (sin 2x) and (cos 2x): [sin 2x cdot cos 2x - 1 (2 sin x cos x)(2 cos^2 x - 1) - 1] Simplify further: [sin 2x cdot cos 2x - 1 (2 sin x cos x)(2 cos^2 x - 1) - 1 2 sin x cos x cdot (2 cos^2 x - 1) - 1] Now we have the simplified forms for numerator and denominator. Let's rewrite the original expression using these simplified forms: [frac{sin 2x - cos 2x - 1}{sin 2x cdot cos 2x - 1} frac{2 sin x cos x - 2 cos^2 x}{2 sin x cos^2 x - 2 sin^2 x}] Factor out (-1) and simplify the expression further: [frac{2 sin x cos x - 2 cos^2 x}{2 sin x cos x - 2 sin^2 x} frac{2 cos x (sin x - cos x)}{2 sin x (cos x - sin x)}] Notice that (cos x - sin x -(sin x - cos x)), so the expression simplifies to: [frac{2 cos x (sin x - cos x)}{2 sin x (cos x - sin x)} frac{cos x (sin x - cos x)}{-sin x (sin x - cos x)}] The ((sin x - cos x)) terms cancel out, leaving us with: [frac{cos x}{-sin x} -cot x] Therefore, the simplified form of the given expression is (-cot x).