Trigonometric identities form a cornerstone of higher mathematics. They encapsulate all the trigonometric ratios and relationships in a framework that enhances the solving of equations and understanding of geometric and algebraic concepts.

Trigonometric identities encompass a wide range of formulas, but people generally group them into categories based on their specific applications and forms.

There are three main categories comprising eight fundamental trigonometric identities. These categories include reciprocal identities, Pythagorean identities and quotient identities.

### Reciprocal Identities

These identities express the basic trigonometric functions in terms of their reciprocal functions:

**Sine and cosecant**: csc(*θ*) = 1/sin(*θ*)**Cosine and secant**: sec(*θ*) = 1/cos(*θ*)**Tangent and cotangent**: cot(*θ*) = 1/tan(*θ*)

### Pythagorean Identities

The Pythagorean trigonometric identities stem from the Pythagorean theorem, also known as the Pythagoras theorem, after the Greek scholar who came up with the mathematical statement.

The trig identities based on the Pythagorean theorem are fundamental to connecting the squares of the primary trigonometric functions:

**Basic Pythagorean identity**: sin^{2}(*θ*) + cos^{2}(*θ*) = 1**Derived for tangent**: 1 + tan^{2}(*θ*) = sec^{2}(*θ*)**Derived for cotangent**: cot^{2}(*θ*) + 1 = csc^{2}(*θ*)

### Quotient Identities

These identities relate the functions through division:

**Tangent as a quotient**: tan(*θ*) = sin(*θ*)/cos(*θ*)**Cotangent as a quotient**: cot(*θ*) = cos(*θ*)/sin(*θ*)

Of course, there are many more trigonometric identities beyond just these core identities that have applications in specific scenarios, such as double angle, triple angle, half angle and sum and difference identities.