Solving the Equation sin2x cos2x - 2sin x cos x 0: A Comprehensive Guide

Solving trigonometric equations is a fundamental skill in mathematics, often used in various fields such as physics, engineering, and signal processing. In this article, we will explore how to solve the trigonometric equation sin2x cos2x - 2sin x cos x 0. This equation will be analyzed step-by-step with detailed explanations and relevant keywords.

Step-by-Step Solution

Let's begin by simplifying the given equation:

Simplification and Rearrangement

The equation sin2x cos2x - 2sin x cos x 0 can be rewritten by expressing 1 as sin2x cos2x and using the double angle identity for sine, sin 2x 2sin x cos x.

1 sin2x cos2x 1 - (sin2x cos2x) -2sin x cos x -2sin x cos x 0 implies sin x cos x 0 or sin x cos x -1

Solving the Simplified Equation

Let's solve the two cases separately:

Case 1: sin x cos x 0

If sin x cos x 0, then either sin x 0 or cos x 0.

sin x 0 implies x nπ cos x 0 implies x (2n 1)π/2

Case 2: sin x cos x -1

The equation sin x cos x -1 is more complex. We can use the identity sin 2x 2sin x cos x to simplify this:

sin 2x -2 implies sin 2x -1. The general solution to this equation is:

2x 2nπ - π/2 implies x nπ - π/4

Further Simplifications and Validation

Let's validate and further simplify the equation to ensure we have a complete set of solutions:

Squaring both sides of the equation, we obtain:

1 - 2sin 2x sin22x 4sin2x 4cos2x 8sin x cos x

This simplifies to:

sin22x - 2sin 2x - 3 0 implies sin 2x - 3sin 2x - 1 0

The first possibility is:

sin 2x 3 is not valid as sin 2x must be in the range [-1, 1].

The second possibility is:

sin 2x -1 implies 2x 2nπ - π/2 implies x nπ - π/4

Conclusion

In summary, the general solution to the equation sin2x cos2x - 2sin x cos x 0 is given by:

x nπ x (2n 1)π/2 x nπ - π/4

These solutions cover all possible values of x for any integer n. Understanding these solutions is crucial for solving similar trigonometric equations and for applications in various fields.

Keywords

trigonometric equations, general solution, sinusoidal functions