Plane trigonometry, by S.L. Loney.

[Exs. LXIV.] EXAMPLES. 435
14. If n be even, prove that
n-1 wr 3w. 5wr n-1
22 sin 2 sin 2 sin 2...... sin 2T r= 1
~ —1 Tn IT n=2 2 COS COS...... cos r.
2n 2n 2n
15. If n be odd, prove that
n- 27r 47r n-1 n1 w n-2
22 sin 2- sin 2*... sin 2 r /n - 2 - os 2 cos.. cos 2n r,
2n n n 2n cos T
and that
l-i w. 3w n-2 27r 4r n- 1
2 2 sin sin-.. sin 7r=- 1=2 2 cos - cos 4-... cos - 7r.
2n 2n 2 2 2n n 2n
sinsr 2r n-1 n
16. Prove that si-sin....sin --- nr
n n n 2n17. If n be odd, prove that
7r 27r 3wr 2(n2 -1)
tan - tan -tan...... tan = n.
n n n n
18. Shew that cos nO
=2n-1(cos 0 cos 7 cos - cos... cosO - cos. 7).
37r (n 2ni
Prove that
19. sin n0=2-1 sin 0 sin ( ) +......sin (+ + — r
r=n- -?'r
= 2n-1 II sin +-.
r=O n
[Put x=l, and 0=2b, in the equation of Art. 362.]
20. cos nq-2-1 sin ( -) sin (0+...... sin 0 + 2- 7r.
[Change f into + 2- in the formula of the preceding question.]
2 n1
21. 2n~l cOs CGOS (0 +n) COS (0 + -)......COS (- 0 + ---71)
= (- 1)2 sin n0, when n is even,
n-l
and = (- 1) 2 cos n0, when n is odd.
Change 0 into + r in the result of Ex. 19.
28-2