Plane trigonometry, by S.L. Loney.

184 TRIGONOMETRY.
C C A B
2 cot 2 2 cos 2 cos 2 cos 2
2 2 2 2
Aso B. C B. A A B]
tan - + tan - sin sin cos - +cos 2 sin
2 2 2L 2 2
A B C A B
2 cos cos cos 2 cos cos -
= -. C.A+B -..... (Art. 69.)
C. A7B C
smn sin sl 2
2 2
We have therefore
2 cot C
a+b+c 2
c A B
tan + tan -
2 2
so that (a + +c) (tan + tan ) = 2c cot-.
2 2j 2
Ex. 4. If the sides of a triangle be in Arithmetical Progression, prove
that so also are the cotangents of half the angles.
W e have given that a+c=2b....................................(1),
and we have to prove that
cot +cot cot......... (2)
2 2 2.(2).
Now (2) is true if
/ ss -a) s (s-c) =2 s(s - b)
(s -b) (s-c) (s -a) (s- b) / (s-c) ( -a) '
or, by multiplying both sides by
(s - a) (s - b) (s - c)
if (s - a)+(s - c)=2 (s - b),
i.e. if 2s- (a + c)=2s - 2b,
i.e. if a+ c=2b, which is relation (1).
Hence if relation (1) be true, so also is relation (2).