Plane trigonometry, by S.L. Loney.

SUBMULTIPLE. ANGLES. 111
Submultiple angles.
109. Since the relations of Art. 105 are true for all
values of the angle A, they will be true if instead of A
A
we substitute -, and therefore if instead of 2A we put
A
2. i.e. A.
2'
Hence we have the relations
A A
sin A = 2 sin cos........... (1)
2..
A A
cos A = cos2 - sin2
_ 2 cos2 1 = 1 -- 2 sing A
2 2
COS2 A_ 2 sin2A. (2),
2 tan A
and tan A = (3).
1 -tan2 a
From (1) we also have
A A
2 sin - cos -
sin A = A A,A ^A
cos2 - + sin -2
2 2
A
2 tan -
=- aaA' by dividing numera1+ tan' 2
A
tor and denominator by cos'.
2t