Verify that each equation is an identity.
sin² β (1 + cot² β) = 1

Verified step by step guidance

1. Start by recalling the Pythagorean identity: \(1 + \cot^2 \beta = \csc^2 \beta\).
2. Substitute \(1 + \cot^2 \beta\) with \(\csc^2 \beta\) in the given equation: \(\sin^2 \beta \cdot \csc^2 \beta = 1\).
3. Remember that \(\csc \beta = \frac{1}{\sin \beta}\), so \(\csc^2 \beta = \frac{1}{\sin^2 \beta}\).
4. Substitute \(\csc^2 \beta\) with \(\frac{1}{\sin^2 \beta}\) in the equation: \(\sin^2 \beta \cdot \frac{1}{\sin^2 \beta} = 1\).
5. Simplify the expression: \(\sin^2 \beta \cdot \frac{1}{\sin^2 \beta} = 1\), which confirms the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity states that for any angle β, sin² β + cos² β = 1. This fundamental relationship between sine and cosine is crucial for verifying trigonometric identities, as it allows us to express one function in terms of another, simplifying the equation.

Cotangent Function

The cotangent function, defined as cot β = cos β / sin β, is the reciprocal of the tangent function. The identity cot² β = cos² β / sin² β is essential for manipulating equations involving cotangent, particularly when combined with the Pythagorean identity to simplify expressions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variable where both sides are defined. Common identities include reciprocal, quotient, and Pythagorean identities. Understanding these identities is key to verifying equations, as they provide the necessary tools to transform and simplify expressions.