Plane trigonometry, by S.L. Loney.

PRODUCT FORMULAE.

On making these substitutions the relations (1) to (4)
become, for all values of C and D,
C+D C-D
sin C + sin D = 2 sin 2 cos.......I,
2 2
C+D C-D
sin C - sin D =2 cos 2 sin 2....II,
C+D C-D
cos C cosD cos cos cos... III,
C+D C -D
and cos D - cos C = 2 sin sin. IV:
2 2
[The student should carefully notice that the left-hand
member of IV is cos D - cos C and not cos C - cos D.]
95. These relations I to IV are extremely important
and should be very carefully committed to memory.
On account of their great importance we give a geometrical proof for the case when C and D are acute angles.
Let AOC be the angle C and A OD the angle D.
Bisect the angle COD by the straight line OE. On OE
take a point P and draw QPR perpendicular to OP to
meet OC and OD in Q and R respectively.
Draw PL, QM and RN perpendicular to OA, and
through R draw RST perpendicular to PL or QM to
meet them in S and T respectively.
Since the angle DOC is C-D, each of the angles
C-D
DOE and EOC is D, and also
2'
C-D C+D
z AOE= AOD +DOE = D+.=
2 2
Since the two triangles POR and POQ are equal in
all respects we have OQ = OR, and PR = PQ, so that
RQ = 2RP.