Trigonometric Identities

.
In mathematics, an "identity" is an equation which is always true. These can be "trivially" true, like "x = x" or usefully true, such as the Pythagorean Theorem's "a² + b² = c²" for right triangles. There are loads of trigonometric identities, but the following are the ones you're most likely to see and use

Basic and Pythagorean Identities

  Notice how a "co-(something)" trig ratio is always the reciprocal of some "non-co" ratio. You can use this fact to help you keep straight that cosecant goes with sine and secant goes with cosine.

sin²(t) + cos²(t) = 1       tan²(t) + 1 = sec²(t)       1 + cot²(t) = csc²(t)

The above, because they involve squaring and the number 1, are the "Pythagorean" identities. You can see this clearly if you consider the unit circle, where sin(t) = y, cos(t) = x, and the hypotenuse is 1.

sin(–t) = –sin(t)       cos(–t) = cos(t)       tan(–t) = –tan(t)

Notice in particular that sine and tangent are odd functions , while cosine is an even function
 Angle Sum and -Difference Identities

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
sin(α – β) = sin(α)cos(β) – cos(α)sin(β)
cos(α + β) = cos(α)cos(β) – sin(α)sin(β)
cos(α – β) = cos(α)cos(β) + sin(α)sin(β)
 

Double Angle Identities

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos²(x) – sin²(x) = 1 – 2sin²(x) = 2cos²(x) – 1

Half Angle Identities

The above identities can be re stated as:

sin²(x) = ½[1 – cos(2x)]

cos²(x) = ½[1 + cos(2x)]

Sum Identities

  Product Identities