Trigonometric Iden tities
You should know and be able to use each of
the following trig identities.
Odd-ev en iden tities:
sin(−x) = −sin x
cos(−x) = cos x
tan (−x) = −tan x
Co-function iden tities:
sin
π
2−x
= cos x
cos
π
2−x
= sin x
tan
π
2−x
= cot x
Pythagorean iden tities:
sin2x+ cos2x= 1
Fron this follow
sec2x= 1 + tan 2x
csc2x= 1 + cot2x
Addition form ulas:
sin(A+B) = sin Acos B+ sin Bcos A
sin(A−B) = sin Acos B−sin Bcos A
cos(A+B) = cos Acos B−sin Asin B
cos(A−B) = cos Acos B+ sin Asin B
tan (A+B) = tan A+ tan B
1−tan Atan B
tan (A−B) = tan A−tan B
1 + tan Atan B
Many useful identities follow from the addi-
tion formulas:
Double-angle form ulas: setting A=B
sin 2θ= 2 sin θcos θ
cos 2θ= cos2θ−sin2θ
cos 2θ= 2cos2θ−1
cos 2θ= 1 −2 sin2θ
Alternativ e very useful versions of the last two are
cos2x=1
2
cos 2x+ 1
sin2x=1
2
1−cos 2x
Pro duct form ulas:
sin xcosy=1
2
sin(x+y) + sin(x−y)
cos xsin y=1
2
sin(x+y)−sin(x−y)
cos xcosy=1
2
cos(x+y) + cos(x−y)
sin xsin y=−1
2
cos(x+y)−cos(x−y)
1
