Verify that each equation is an identity.
sin(x + y)/cos(x - y) = (cot x + cot y)/(1 + cot x cot y)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, provided the expressions are defined. Common identities include the Pythagorean identities, angle sum and difference identities, and reciprocal identities. Understanding these identities is crucial for verifying equations and simplifying trigonometric expressions.

Angle Sum and Difference Formulas

The angle sum and difference formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For example, sin(x + y) = sin x cos y + cos x sin y. These formulas are essential for manipulating and simplifying expressions involving sums or differences of angles in trigonometric equations.

Cotangent Function

The cotangent function, defined as cot(x) = cos(x)/sin(x), is the reciprocal of the tangent function. It plays a significant role in trigonometric identities and can be expressed in terms of sine and cosine. Understanding how to manipulate cotangent, along with its relationships to other trigonometric functions, is vital for verifying identities involving cotangent.