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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.

We will also learn how the different Trigonometric identities are derived and how they may be used.

Related Topics: More Lessons on Trigonometry

sin (*a* + *b*) = sin *a* cos *b* + cos *a* sin *b*

cos (*a* + *b*) = cos *a* cos *b* – sin *a* sin *b*

sin (*a* - *b*) = sin *a* cos *b* - cos *a *sin *b*

cos (*a* - *b*) = cos *a* cos *b* + sin *a* sin *b*

sin 2*a* = 2 sin *a* cos *a*

cos 2*a* = cos ^{2}*a* – sin ^{2}*a* = 2 cos^{2}*a* – 1 = 1 – 2 sin^{2}*a*

This video shows how to derive the Quotient, Reciprocal and Pythagorean identities.

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 (

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.

How to Prove the Addition and Subtraction Formula for sin(x+y) or sin(x-y)

Using the Sine and Cosine Addition

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

How to use the sine and cosine subtraction formulas to prove the cofunction identities.
How to use the sine and cosine addition formulas to prove the cofunction identities.
## Sum and Difference Identities for Tangent

## Double Angle Identities

Double Angle Formulas

The double angles sin(2theta) and cos(2theta) can be rewritten as sin(theta+theta) and cos(theta+theta). Applying the cosine and sine addition formulas, we find that sin(2theta)=2sin(theta)cos(theta) and cos(2theta)=cos^{2}(theta) - sin^{2}(theta).cos(2theta)=cos^{2}(theta) - sin^{2}(theta). Combining this formula with the Pythagorean Identity, cos^{2}(theta) + sin^{2}(theta) = 1, two other forms appear: cos(2theta) = 2cos^{2}(theta) -1 and cos(2theta) = 1 - 2sin^{2}(theta).

How to use the sine and cosine addition formulas to prove the double-angle formulas.

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

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

How to use the sine and cosine subtraction formulas to prove the cofunction identities.

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

How to use the sine and cosine addition formulas to prove the double-angle formulas.

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

How to use the double-angle identities to determine function values.

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.

How to use the half-angle identities to determine function values.

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