Plane trigonometry, by S.L. Loney.

CHAPTER VII.

TRIGONOMETRICAL RATIOS OF THE SUM AND DIFFERENCE
OF TWO ANGLES.
88. Theorem. To prove that
sin (A + B) = sin A cos B + cos A sin B,
and cos (A + B) = cos A cosB - sin A sin B.

P
B
/R
M A
0 MQA

C

A' M U Q

A

Let the revolving line start from OA and trace out
the angle AOB (=A), and then trace out the further
angle BOC (= B).
In the final position of the revolving line take any
point P, and draw PM and PNT perpendicular to OA and
OB respectively; through N draw NR parallel to AO to
meet MP in R and draw NQ perpendicular to OA.