# Trig identities

Trigonometric identities are equations that are used to describe the many relationships that exist between the trigonometric functions. Among other uses, they can be helpful for simplifying trigonometric expressions and equations.

The following shows some of the identities you may encounter in your study of trigonometry.

### Reciprocal identities

sin(θ)·csc(θ) = 1

cos(θ)·sec(θ) = 1

tan(θ)·cot(θ) = 1

### Quotient identities

### Cofunction identities

### Odd/even identities

sin(-θ) = -sin(θ)

cos(-θ) = cos(θ)

tan(-θ) = -tan(θ)

csc(-θ) = -csc(θ)

sec(-θ) = sec(θ)

cot(-θ) = -cot(θ)

### Pythagorean identities

cos

^{2}(θ) + sin^{2}(θ) = 11 + tan

^{2}(θ) = sec^{2}(θ)1 + cot

^{2}(θ) = csc^{2}(θ)Example:

Verify that cos(x)·tan(x) + csc(x)·cos^{2}(x) = csc(x) using trigonometric identities.

cos(x)·tan(x) + csc(x)·cos^{2}(x) | |

= | |

= | |

= | |

= | |

= | csc(x) |

## Trigonometric formulas

There are many formulas used in trigonometry that involve one or more angles or sides of a triangle.

### Sum and difference formulas

sin(x ± y) = sin(x)·cos(y) ± cos(x)·sin(y) |

cos(x ± y) = cos(x)·cos(y) ∓ sin(x)·sin(y) |

### Double angle formulas

sin(2θ) = 2·sin(θ)·cos(θ) |

cos(2θ) = cos^{2}(θ) - sin^{2}(θ) = 1 - 2·sin^{2}(θ) = 2·cos^{2}(θ) - 1 |

### Half angle formulas

## Less frequently used identities

Though the identities below are not used as frequently as some of those above, you may still come across them in your studies.

### Product to sum identities

2·sin(x)·cos(y) = sin(x+y) + sin(x-y) |

2·cos(x)·sin(y) = sin(x+y) - sin(x-y) |

2·cos(x)·cos(y) = cos(x+y) + cos(x-y) |

2·sin(x)·sin(y) = cos(x-y) - cos(x+y) |

### Sum to product identities

Example:

Find the exact value of cos(105°) + cos(15°) using a sum to product identity: