Mathematics reference
Trigonometric identities
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Various identities and properties essential in trigonometry.

Legend.
  • x and y are independent variables,
  • d is the differential operator,
  • int is the integration operator,
  • C is the constant of integration.
Identities.
tan x = sin x/cos x equation 1
cot x = cos x/sin x equation 2
sec x = 1/cos x equation 3
csc x = 1/sin x equation 4
cot x = 1/tan x equation 5
sin2 x + cos2 x = 1 equation 6
tan2 x + 1 = sec2 x equation 7
1 + cot2 x = csc2 x equation 8
cos (x +- y) = cos x cos y -+ sin x sin y equation 9
sin (x +- y) = sin x cos y +- cos x sin y equation 10
cos (-x) = cos x equation 11
cos (pi/2 - x) = sin x equation 12
cos (2 x) = cos2 x - sin2 x = 2 cos2 x - 1 = 1 - 2 sin2 x equation 13
cos (x/2) = +-[(1/2) (1 + cos x)]1/2 equation 14
sin (-x) = -sin x equation 15
sin (pi/2 - x) = cos x equation 16
sin (2 x) = 2 sin x cos x equation 17
tan (x +- y) = (tan x +- tan y)/(1 -+ tan x tan y) equation 18
tan (-x) = -tan x equation 19
tan (2 x) = 2 tan x/(1 - tan2 x) equation 20
tan (x/2) = sin x/(1 + cos x) equation 21
cos (pi +- x) = -cos x equation 22
sin (pi +- x) = -+sin x equation 23
tan (pi +- x) = +-tan x equation 24
Derivatives.
(d/dx) sin x = cos x equation 25
(d/dx) cos x = -sin x equation 26
(d/dx) tan x = sec2 x equation 27
(d/dx) cot x = -csc2 x equation 28
(d/dx) sec x = sec x tan x equation 29
(d/dx) csc x = -csc x cot x equation 30
Integrals.
int sin x dx = -cos x + C equation 31
int cos x dx = sin x + C equation 32
int tan x dx = -ln |cos x| + C equation 33
int cot x dx = ln |sin x| + C equation 34
int sec x dx = ln |sec x + tan x| + C equation 35
int csc x dx = -ln |csc x + cot x| + C equation 36
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