# Formulas and Tables

Trigonometry# Basic Trigonometric Identities

Trigonometric functions: \(\sin \alpha,\) \(\cos \alpha,\) \(\tan \alpha,\) \(\cot \alpha,\) \(\sec \alpha,\) \(\csc \alpha\)

Set of integers: \(\mathbb{Z}\)

Integers: \(n\)

- Trigonometric identities establish a connection between trigonometric functions of the same argument (angle \(\alpha\)).
- Pythagorean trigonometric identity

\({\sin ^2}\alpha + {\cos ^2}\alpha = 1\)

This identity is the result of application of the Pythagorean theorem to a triangle in the unit circle. - Relationship between the cosine and tangent

\(\large{\frac{1}{{\cos^2}\alpha}}\normalsize – {\tan ^2}\alpha = 1\) or \(\sec^2\alpha – {\tan ^2}\alpha = 1.\)

This identity follows from the Pythagorean trigonometric identity and is obtained by dividing the left and right sides by \(\cos^2 \alpha\). It is assumed that \(\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,\) \(n \in \mathbb{Z}\). - Relationship between the sine and cotangent

\(\large{\frac{1}{{\sin^2}\alpha}}\normalsize – {\cot ^2}\alpha = 1\) or \(\csc^2\alpha – {\cot ^2}\alpha = 1.\)

This formula also follows from the Pythagorean trigonometric identity (it is obtained by dividing the left and right sides by \(\sin^2 \alpha\). It is assumed that \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}\). - Definition of tangent

\(\tan \alpha = \large{\frac{\sin \alpha}{\cos \alpha}}\normalsize ,\) where \(\alpha \ne \large{\frac{\pi}{2}}\normalsize + \pi n,\) \(n \in \mathbb{Z}\). - Definition of cotangent

\(\cot \alpha = \large{\frac{\cos \alpha}{\sin \alpha}}\normalsize ,\) where \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}\). - Consequence of the definitions of tangent and cotangent

\(\tan \alpha \cdot \cot \alpha = 1,\) where \(\alpha \ne \large{\frac{\pi n}{2}}\normalsize ,\) \(n \in \mathbb{Z}.\) - Definition of secant

\(\sec \alpha = \large{\frac{1}{\cos \alpha}}\normalsize,\) \(\alpha \ne \large{\frac{\pi}{2}}\normalsize +\pi n,\) \(n \in \mathbb{Z}.\) - Definition of cosecant

\(\csc \alpha = \large{\frac{1}{\sin \alpha}}\normalsize,\) \(\alpha \ne \pi n,\) \(n \in \mathbb{Z}.\)

### Related Pages

- Definition and Graphs of Trigonometric Functions
- Cofunction and Reduction Identities
- Relationships between Trigonometric Functions
- Addition and Subtraction Formulas
- Double and Multiple Angle Formulas
- Half-Angle Formulas
- Sum-to-Product Identities
- Product-to-Sum Identities
- Powers of Trigonometric Functions
- Basic Trigonometric Equations
- Basic Trigonometric Inequalities
- Inverse Trigonometric Functions