Plane trigonometry, by S.L. Loney.

DE MOIVRE'S THEOREM. 313
So
[cos a + V - 1 sin a] [cos8 - 1 sin/3] [cos 7 + - 1 sin y]
= [cos (a + /) + /- 1 sin (a + 3)] [cos 7 + 4 - 1 sin 7]
= [cos (a + 3) cos y - sin (a +-,) sin y]
+4 - 1 [sin (a + 3) cos 7 + cos (a + 43) sin y]
= cos (a +, - 7) + 4- 1 sin (a +/3 + 7).
This process may evidently be continued indefinitely,
so that
[cos a + V -1 sin a] [cos 3 + V/- 1 sin,3][cos 7 + /-1 sin 7]...... to n factors
= cos (a 3 + 7 +... ton terms) + /-l sin [a + 43 +y...
to n terms].
In this expression put
a= / =7=...... =,
so that we have
[cos + /- I sin 8]1 = cos no + J - 1 sin nO.
Case II. Let n be a negative integer and equal to
- m.
We have, by the ordinary law of indices,
(cos 0 + 4- 1 sin )n = (cos 0 + - 1 sin 0)-7"
1 1
(cos 0 + - 1 sin 9)m cos mO+ 4/- - sin m0
by Case I,
cos mO - V~ 1 sin mO
(cos m + / -1 sin mO) (cos mO - V - 1 sin mO)
cos mO - V - i sin m0
= - = cos m0 - V/ —1 sin m0
cos2,MO + sin2 mO
= cos (- m) 0 + V - 1 sin (- m) 0
= cos no + V - 1 sin nO.