The Pythagorean identities are fundamental equations in trigonometry that relate the squares of the sine, cosine, and tangent functions. The most common forms are: \sin^2(\theta) + \cos^2(\theta) = 1 and 1 + \tan^2(\theta) = \sec^2(\theta) . These identities are derived from the Pythagorean theorem, which describes the relationship between the sides of a right triangle. These identities are useful for simplifying trigonometric expressions and solving equations. They help establish connections between different trigonometric functions, making it easier to analyze and understand various properties of triangles and periodic functions in mathematics.