TRIGONOMETRIC sutra

TRIGONOMETRIC IDENTITIES
Reciprocal identities
sin u =
1
csc u
cos u =
1
sec u
tan u =
1
cot u
cot u =
1
tan u
csc u =
1
sin u
sec u =
1
cos u
Pythagorean Identities
sin2 u + cos2 u = 1
1 + tan2 u = sec2 u
1 + cot2 u = csc2 u
Quotient Identities
tan u =
sin u
cos u
cot u =
cos u
sin u
Co-Function Identities
sin(

2
􀀀 u) = cos u cos(

2
􀀀 u) = sin u
tan(

2
􀀀 u) = cot u cot(

2
􀀀 u) = tan u
csc(

2
􀀀 u) = sec u sec(

2
􀀀 u) = csc u
Parity Identities (Even & Odd)
sin(􀀀u) = 􀀀sin u cos(􀀀u) = cos u
tan(􀀀u) = 􀀀tan u cot(􀀀u) = 􀀀cot u
csc(􀀀u) = 􀀀csc u sec(􀀀u) = sec u
Sum & Di erence Formulas
sin(u v) = sin u cos v cos u sin v
cos(u v) = cos u cos v sin u sin v
tan(u v) =
tan u tan v
1 tan u tan v
Double Angle Formulas
sin(2u) = 2 sin u cos u
cos(2u) = cos2 u 􀀀 sin2 u
= 2 cos2 u 􀀀 1
= 1 􀀀 2 sin2 u
tan(2u) =
2 tan u
1 􀀀 tan2 u
Power-Reducing/Half Angle For-
mulas
sin2 u =
1 􀀀 cos(2u)
2
cos2 u =
1 + cos(2u)
2
tan2 u =
1 􀀀 cos(2u)
1 + cos(2u)
Sum-to-Product Formulas
sin u + sin v = 2 sin

u + v
2

cos

u 􀀀 v
2

sin u 􀀀 sin v = 2 cos

u + v
2

sin

u 􀀀 v
2

cos u + cos v = 2 cos

u + v
2

cos

u 􀀀 v
2

cos u 􀀀 cos v = 􀀀2 sin

u + v
2

sin

u 􀀀 v
2

Product-to-Sum Formulas
sin u sin v =
1
2
[cos(u 􀀀 v) 􀀀 cos(u + v)]
cos u cos v =
1
2
[cos(u 􀀀 v) + cos(u + v)]
sin u cos v =
1
2
[sin(u + v) + sin(u 􀀀 v)]
cos u sin v =
1
2
[sin(u + v) 􀀀 sin(u 􀀀 v)]