In Exercises 53–62, solve each equation on the interval [0, 2𝝅). cot x (tan x - 1) = 0

Verified step by step guidance

1. Start by analyzing the given equation: \(\cot x (\tan x - 1) = 0\). Since this is a product equal to zero, use the zero product property which states that if \(AB = 0\), then either \(A = 0\) or \(B = 0\).
2. Set each factor equal to zero separately: 1) \(\cot x = 0\) 2) \(\tan x - 1 = 0\)
3. Solve the first equation \(\cot x = 0\). Recall that \(\cot x = \frac{\cos x}{\sin x}\), so \(\cot x = 0\) when \(\cos x = 0\) (and \(\sin x \neq 0\)). Find all \(x\) in \([0, 2\pi)\) where \(\cos x = 0\).
4. Solve the second equation \(\tan x - 1 = 0\) which simplifies to \(\tan x = 1\). Find all \(x\) in \([0, 2\pi)\) where the tangent of \(x\) equals 1.
5. Combine all solutions from both equations and ensure they lie within the interval \([0, 2\pi)\). These combined values of \(x\) will be the solutions to the original equation.

Key Concepts

Trigonometric Functions and Their Domains

Understanding the definitions and properties of cotangent and tangent functions is essential. Cotangent is the reciprocal of tangent, and both are periodic with period π. Knowing their domains and where they are undefined helps in solving equations and identifying valid solutions within the interval [0, 2π).

Zero-Product Property

The zero-product property states that if a product of two factors equals zero, then at least one of the factors must be zero. This principle allows us to split the equation cot x (tan x - 1) = 0 into two simpler equations: cot x = 0 and tan x - 1 = 0, which can be solved separately.

Solving Trigonometric Equations on a Specific Interval

When solving trigonometric equations, it is important to find all solutions within the given interval, here [0, 2π). This involves finding general solutions using known values of trig functions and then restricting them to the specified domain, considering periodicity and any domain restrictions.