Trigonometric Identities

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.

Quotient Identities

Reciprocal Identities

Pythagorean Identities

Co-function Identities

Sum Identities or Addition Formulas

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

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

Difference Identities or Subtraction Formulas

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

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

Double Angle Formulas

sin 2a = 2 sin a cos a

cos 2a = cos 2a – sin 2a = 2 cos 2a – 1 =  1 – 2 sin 2a

Even-odd Identities

Sum-to-product Formulas

Product-to-sum Formulas

Half Angle Formulas

Videos

The following video describes the quotient, reciprocal and Pythagorean identities.

Cosine Addition Formula
The cosine addition formula calculates the cosine of an angle that is either the sum or difference of two other angles. It arises from the law of cosines and the distance formula. By using the cosine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.

Sine Addition Formula
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.

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(?-x) = -cos(x). The formulas also give the tangent of a difference formula, for tan(alpha-beta).

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). Also, cos(2theta)=cos2(theta) - sin2(theta), see other forms for the two derivations. These results reappear in integral calculus, when remembering them can be the difference between a right and wrong answer.

Other Forms of the Cosine Double-Angle Formula T
he cosine double angle formula is cos(2theta)=cos2(theta) - sin2(theta). Combining this formula with the Pythagorean Identity, cos2(theta) + sin2(theta)=1, two other forms appear: cos(2theta)=2cos2(theta)-1 and cos(2theta)=1-2sin2(theta). These can be used to find the power-reduction formulas, which reduce a second degree or higher trig function to a first degree. These formulas are very useful in Calculus.

Half Angle Identities
The half angle identities come from the power reduction formulas using the key substitution alpha=theta/2 twice, once on the left and right sides of the equation. With half angle identies, on the left side, this yields (after a square root) cos(theta/2) or sin(theta/2); on the right side cos(2theta) becomes cos(?) because 2(1/2)=1. For a problem like sin(pi/12), remember that theta/2=pi/12, or theta=pi/6, when substituting into the identity.

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