Trigonometric identities (trig identities) are equalities that involve trigonometric functions that are true for all values of the occurring variables. These identities are useful when we need to simplify expressions involving trigonometric functions.

The following is a list of useful Trigonometric identities: Quotient Identities, Reciprocal Identities, Pythagorean Identities, Co-function Identities, Addition Formulas, Subtraction Formulas, Double Angle Formulas, Even Odd Identities, Sum-to-product formulas, Product-to-sum formulas.

Scroll down the page to learn how the different Trigonometric identities are derived and how they may be used.This video explains how to use cofunction identities to solve trigonometric equations.

This video shows how to derive an identity for Cosine of a sum of two angles. It will use the unit circle definition for sine and cosine, the Pythagorean Identity, the distance formula between two points, and some algebra to derive an identity for cos (

Examples using the formula are provided.

More examples of using the sum and difference identities to find value other trig values.

Starting with the cofunction identities, the sine addition formula is derived by applying the cosine difference formula. There are two main differences from the cosine formula: (1) the sine addition formula adds both terms, where the cosine addition formula subtracts and the subtraction formula adds; and (2) the sine formulas have sin-sin and cos-cos. Both formulas find values for angles.

Formulas to Prove Identities: Applying the cosine addition and sine addition formulas proves the cofunction, add π, and supplementary angle identities. Using the formulas, we see that sin(π/2-x) = cos(x), cos(π/2-x) = sin(x); that sin(x + π) = -sin(x), cos(x + π) = -cos(x); and that sin(π-x) = sin(x), cos(π-x) = -cos(x). The formulas also give the tangent of a difference formula, for tan(α - β).

The double angles sin 2θ and cos 2θ can be rewritten as sin(θ + θ) and cos(θ + θ). Applying the cosine and sine addition formulas, we find that sin 2θ = 2sin θ cos θ and cos 2θ = cos

The derivation of the double angle identities for sine and cosine, followed by some examples.

The derivations of the half-angle identities for both sine and cosine, plus listing the tangent ones. Then a couple of examples using the identities.

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