Plane trigonometry, by S.L. Loney.

EXAMPLES.

447

377. Ex. Prove that
cosh 2a - cos 20
=2sin201+[l-2 [1+(2 a a)
[ a2 a (a ad
+ r n- 0 + 4(27r+)]1 (2 - -0-.......adinf,
=2sin201[1+ (O+ra)2'
where r is zero or any positive or any negative integer.
We have
cosh 2a - cos 20 = cos 2ai - cos 20= 2 sin (0 + ai) sin (0 - ai)
72 [ 2.,2,.2
'2 ((0+ ai) -- ( ai)] - ( ( -.i) ]
x(0-.ai) [1- 1 ][,- 3......... (1).
Now [I - ( [1 - a
(_ r + + ai)l r- - ( i) [( + 0 - ai) (7r - 0 + ai).2 (.
(7r+0)2+a2 (T r-)2+a2
7r2 ' 2
Hence (1) gives
cosh 2a - cos 20 2 (02 + a2) + 2] [(- ) a2(2 + )2 + 2]
(2r _ )2 + a2] adinf.............(2).
In (2) put a=O and we have
2s = 2.( )2 ( 0)2 (2r+ )2 (27 - 0)2 ad inf.............(3).
7 7 71 7.2 -
Dividing (2 by ) (3) we have
cosh 2a - cos 20
=2sin2o [+ 2 1+ - _)[1( i )L[1+(2)i - a+)i
[1 + (2 )]......ad inf.
into 0 + and they are found to be 2cos2 1+ ( + ) where r is
any odd integer, positive or negative.