Trigonometric Laws for all

In a right-angled triangle,
Sinθ= Opposite Side/Hypotenuse
Cosθ= Adjacent Side/Hypotenuse
Tanθ= Sinθ/Cosθ  = Opposite Side/Adjacent Side
Cosecθ = 1/Sinθ= Hypotenuse/Opposite Side
Secθ = 1/Cosθ = Hypotenuse/Adjacent Side
Cotθ = 1/tanθ = Cosθ/Sinθ = Adjacent Side/Opposite Side
SinθCosecθ = CosθSecθ = TanθCotθ = 1
Sin(90-θ) = Cosθ, Cos(90-θ) = Sinθ
Sin²θ + Cos²θ = 1
Tan²θ + 1 = Sec²θ
Cot²θ + 1 = Cosec²θ

Addition and subtraction formula:-
Sin(A+B) = SinACosB + CosASinB
Sin(A-B) = SinACosb - CosASinB
Cos(A+B) = CosACosB - SinASinB
Cos(A-B) = CosACosB + SinASinB
Tan(A+B) = (TanA+TanB)/(1-TanATanB)
Tan(A-B) = (TanA - TanB)/(1+TanATanB)
Cot (A+B) = (CotACotB-1)/(CotA + CotB)
Cot(A-B) = (CotACotB+1)/(CotB-CotA)
Sin(A+B)+Sin(A-B) = 2SinACosB
Sin(A+B)-Sin(A-B) = 2CosASinB
Cos(A+B)+Cos(A-B) = 2CosACosB
Edited : Cos(A - B) - Cos(A + B) = 2SinASinB
SinC + SinD = 2Sin[(C+D)/2]Cos[(C-D)/2]
SinC - SinD = 2Cos[(C+D)/2]Sin[(C-D)/2]
CosC + CosD = 2Cos[(C+D)/2]Cos[(C-D)/2]
CosC - CosD = 2Sin[(C+D)/2]Sin[(D-C)/2]
Sin2θ = 2SinθCosθ = (2tanθ)/(1+tan²θ)
Cos2θ = Cos²θ - Sin²θ = 2Cos²θ - 1= 1 - 2Sin²θ =
(1-tan²θ)/(1+tan²θ)
Tan2θ = 2tan θ/(1-tan²θ)

(Angles are given in degrees, 90 degrees, 180 degrees etc.)
I.
Sin(-θ)=-Sinθ
Cos(-θ) = Cosθ
tan(-θ) = -tanθ
cot(-θ) = -cotθ
sec(-θ) = secθ
cosec(-θ)= - cosecθ
II.
sin(90-θ) = cosθ
cos(90-θ) = sinθ
tan(90-θ) = cotθ
cot(90-θ) = tanθ
sec(90-θ) = cosecθ
cosec(90-θ) = secθ
III.
sin(90+θ) = cosθ
cos(90+θ) = -sinθ
tan(90+θ) = -cotθ
cot(90+θ) = -tanθ
sec(90+θ) = -cosecθ
cosec(90+θ) = secθ
IV.
sin(180-θ) = sinθ
cos(180-θ) = -cosθ
tan(180-θ) = -tanθ
cot(180-θ) = cotθ
sec(180-θ) = -secθ
cosec(180-θ) = cosecθ
V.
sin(180+θ) = -sinθ
cos(180+θ) = -cosθ
tan(180+θ) = tanθ
cot(180+θ) = cotθ
sec(180+θ) = -secθ
cosec(180+θ) = -cosecθ

Formulas which express the sum or difference in product

Formulae which express products as sums or difference of Sines and Cosines

Trignometric ratios of Multiple Angles

Trignometric ratios of 3θ

Trignometric ratios of sub-multiple angles

Properties of Inverse Trignometric Functions

Properties of Triangles
Sine Formula (or  Law of Sines)
In any ΔABC,

Cosine  Formula (or Law of Cosines)
In any ΔABC,

These  formulas are also written as

Projection formulas
In any ΔABC,

Half-Angles and Sides

In any ΔABC,

Area of  a Triangle

Hero's fromula

Incircle and Circumcircle
A circle which touches the three sides of a traingle internally is called the incircle.The center of the circle is called the incentre and the raidus is called the inradius.
If r is the inradius, then

The  circle which passes through the vertices of a triangle is called the circumcircle of a triangle or circumscribing circle. The centre of this circle is the circumcentre and the radius of the circumcircle is the circumradius.
If R is the circumradius, then

If Δ is the area of the triangle,

Hyperbolic Functions

Relation between circular and hyperbolic functions

Addition formulas for Hyperbolic functions

Periods of hyperbolic functions

Inverse Hyperbolic functions

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