Trigonometric Identities Purplemath

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Trigonometric Identities
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.
The following (particularly the first of the three below) are called "Pythagorean" identities.
sin2(t) + cos2(t) = 1
tan2(t) + 1 = sec2(t)
1 + cot2(t) = csc2(t)
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 & Pythagorean, Angle-Sum & -Difference, Double-Angle, Half-Angle, Sum, Product
csc(x) = 
sin(x)
1
sin(x) = 
csc(x)
1
sec(x) = cos(x)
1
cos(x) = sec(x)
1
cot(x) = = tan(x)
1
sin(x)
cos(x)
tan(x) = = cot(x)
1
cos(x)
sin(x)
Note that the three identities above all involve squaring and the number 1. You can see the
Pythagorean-Thereom relationship clearly if you consider the unit circle (unitcirc.htm), where the angle is
t, the "opposite" side is sin(t) = y, the "adjacent" side is cos(t) = x, and the hypotenuse is 1.
We have additional identities related to the functional status of the trig ratios:
sin(–t) = –sin(t)
cos(–t) = cos(t)
tan(–t) = –tan(t)
Notice in particular that sine and tangent are odd functions (fcnnot3.htm), being symmetric about the
origin, while cosine is an even function (fcnnot3.htm), being symmetric about the y-axis. The fact that
you can take the argument's "minus" sign outside (for sine and tangent) or eliminate it entirely
(for cosine) can be helpful when working with complicated expressions.
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)
By the way, in the above identities, the angles are denoted by Greek letters (grklttrs.htm). The a-type
letter, "α", is called "alpha", which is pronounced "AL-fuh". The b-type letter, "β", is called "beta", which is
pronounced "BAY-tuh".
Double-Angle Identities
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2(x) – sin2(x) = 1 – 2 sin2(x) = 2 cos2(x) – 1
Half-Angle Identities
tan(α + β) = 1 − tan(α) tan(β)
tan(α) + tan(β)
tan(α − β) = 
1 + tan(α) tan(β)
tan(α) − tan(β)
tan(2x) = 
1 − tan (x)2
2 tan(x)
sin = ± ( 2
x
) 2
1 − cos(x)
The above identities can be re-stated by squaring each side and doubling all of the angle measures. The
results are as follows:
Sum Identities
Product Identities
cos = ± ( 2
x) 2
1 + cos(x)
tan = ± ( 2
x) 1 + cos(x)
1 − cos(x)
= sin(x)
1 − cos(x)
= 1 + cos(x)
sin(x)
sin (x) = [1 − cos(2x)]2 2
1
cos (x) = [1 + cos(2x)]2 2
1
tan (x) = 2
1 + cos(2x)
1 − cos(2x)
sin(x) + sin(y) = 2 sin cos ( 2
x + y) ( 2
x − y)
sin(x) − sin(y) = 2 cos sin (
2
x + y) (
2
x − y)
cos(x) + cos(y) = 2 cos cos ( 2
x + y) ( 2
x − y)
cos(x) − cos(y) = −2 sin sin (
2
x + y) (
2
x − y)
sin(x) cos(y) = [ sin(x + y) + sin(x − y)]2
1
cos(x) sin(y) = [ sin(x + y) − sin(x − y)]2
1
cos(x) cos(y) = [ cos(x − y) + cos(x + y)]2
1
sin(x) sin(y) = [ cos(x − y) − cos(x + y)]2
1
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You will be using all of these identities, or nearly so, for proving other trig identities and for solving trig
equations. However, if you're going on to study calculus, pay particular attention to the restated sine and
cosine half-angle identities, because you'll be using them a lot in integral calculus.
URL: http://www.purplemath.com/modules/idents.htm