Fundamental Law of Trigonometry
Let us assume that α > β > 0.
Consider a unit circle with centre O.
Let the terminal side of angles α and β cut the unit circle at A and B respectively.
Evidently AOB = α - β.
Take a point C on the unit circle so that XOC = AOB = α - β.
Join A with B and C with D.
The coordinates of A are ( cos α , sin α ).
The coordinates of B are ( cos β , sin β ).
The coordinates of C are [ cos(α - β) , sin(α - β) ].
The coordinates of D are ( 1 , 0 ) Now AOB and COD are congruent.