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Trigonometry
 The addition formulas and related identities
 The sum and difference formulas for the trigonometric functions

 Deriving the addition formulas for sine and cosine functions

Coordinates of the terminal point P(cos (a + b), sin (a + b)) of the arc a + b on the unit circle shown on the right figure, we can write as
sin (a + b) = PxP = u + v    (1)
cos (a + b) = OPx = m - n   (2)

From the right triangles, OBP, CBP and OAB it follows that,
in DOBP,    OB = cos b  and  BP = sin b

after substituting the obtained values for, u, v, m and n into (1) and (2)

sin (a + b) = sin a cos b + cosa sin b
cos (a + b) = cos a cos b - sin a sin b

By replacing b with - b in the above identities, we get

sin [a + (-b)] = sin a · cos (-b) + cosa · sin (-b)     and since    cos (-b) = cos b
cos [a + (-b)] = cos a · cos (-b) - sin a · sin (-b)       and        sin (-b) = - sin b
                         therefore,        sin (a - b) = sin a cos b - cosa sin b
cos (a - b) = cos a cos b + sin a sin b
  The addition formulas for tangent and cotangent functions

The addition formulas for the tangent and cotangent functions we derive from the definitions, thus

and by dividing the numerator and denominator by  cos a cos b,
Using the relation
and after dividing the numerator and denominator by  sin a sin b,
Replacing  b with - b and substituting  tan (- b) = - tan b  and  cot (- b) = - cot b,  we get
 Trigonometric functions of double angles, double angle formulas
By substituting b with a in the sum formulas,
sin (a + b) = sin a cos b + cosa sin b      and     cos (a + b) = cos a cos b - sin a sin b,
and
we get for example,   sin 2a = sin (a + a) = sin a cos a + cosa sin a = 2sin a cos a,   or 
sin 2a = 2 sin a cos a        and        cos 2a = cos2 a - sin2 a

The double angle formula for the cosine function can be expressed by sine or cosine function using the      identity sin2 a + cos2 a = 1,
cos 2a = 2cos2 a - 1            or            cos 2a = 1 - 2sin2 a
             and          1 + cos 2a = 2cos2 a             or            1 - cos 2a = 2sin2 a
 Trigonometric functions expressed by the half angle
Substituting a/2 in double angle formulas we obtain trigonometric functions expressed by the half angle,
 Trigonometric functions of double angles expressed by the tangent function
The double angle formulas can be expressed in terms of a single function, so using the identity
and dividing the numerator and denominator by cos2 a we get
and by using
                          dividing the right side by cos2 a obtained is
then
 Trigonometric functions expressed by the tangent of the half angle
Replacing a by a/2 in the above identities, we get
 Half angle formulas
Using the identities in which trigonometric functions are expressed by the half angle,
and applying the definitions of the functions, tangent and cotangent
  Trigonometric functions expressed by the cosine of the double angle
Replacing a/2 by a in the above identities, we get
 
 
 
 
 
 
 
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