Plane trigonometry, by S.L. Loney.

[Exs. LXV.] INFINITE PRODUCTS. EXAMPLES.

449

1 1 1 1 1
10. sec2 0 (7 r 20)2 + 2 + (3 - 2)2+ (3+ 2)2 ad inf.
[Apply the process of Art. 376 to the result obtained.]
I 1 1 1 1
11. cosec + ( + ( + (2 + ( +... ad inf.
Prove that
12. sin (a - ) (1
sin a a/ \ r-a ) + a)
(1+2 ) - a )(1 2 ) a -)
= 1 (1- ), where r is any positive or negative integer or zero.
13. sin (al) = 1+ ---, where r is any positive or negative
sin a- a+rrTr/
integer, including zero.
cos(a+0) 1 — 20 \ f 2_( 0 \ + 20 20
cosa \ 7+2aJ \ 7r-2aJ \ 37r+2ay k 3r-2a
= H 1+ 2+ --, where r is any odd integer positive or negative.
cos (a - 0) _ r- 20.
15. -os a L=[1 2a + r,,where r is any odd integer, positive
or negative.
cos +cosa F 02 _____ 02 )2]
O 02]
C6 s+ecosa - 1- 0+ a O (,l _ — a)2
L- 1(r7r + a)
C1 (83r- a)....."""
1 — H 1. a '
where r is any odd integer positive or negative.
[Multiply together the results of Exs. 14 and 15 and then change 20
and 2a into 0 and a.]
cos 0- cos a 02
17. 1-co 1 2 — l( )
1 - cos a a 2 (2wr + a)y
i (2 - a)2 (4r + a)2......
[ (a + r7r)2] 2
where r is any even positive or negative integer, including zero.
Hence deduce the factors of cosh x - cos a.
L. T. 29