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

The addition formulas and related identities
The sum and difference formulas for the trigonometric functions - Trigonometric functions of the sum or difference of two angles in terms of separate functions of the angles.
Deriving the addition formulas for sine and cosine functions
 The 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,
By replacing b with -b in the above identities, and substituting
tan (-b) = - tan b  and  cot (-b) = - cot b,  we get
Trigonometry contents A
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