Plane trigonometry, by S.L. Loney.

TANGENT OF THE SUM OF ANGLES.

133

81 -S3 + 8 - S7 + -*+ tan A,,+,
1-s3+ s8...
S1- + Ss * *.
- s 3 + s...tan A,,+
1 -- S2 + S4...
Let tan Al, tan A,... tan A,,l+ be respectively called
ti, to,... t+4i.
Then tan (A1 + A2 +... + A1+1)
(S-S + S5, *...) + t+l ( -S2 +,...)
(1 - S+ S4...) - (s - S, + 5s...)tn+
(S1 + t+l) - (83 + S2 tl+1) + (S5 + S4 t+1l)...
1 - (2 + S tn+1) + (S4 + S3 t+1) - (s6 + S5 tl+1)...
But si + t,+-1 = (t, + t2 -... to) + t,+l
= the sum of the (n + 1) tangents,
S2 + S1 t2+fl = (tt + 3...) + t2 + + tn) tn4+1
= the sum, two at a time, of the (n + 1) tangents.
83 + s2 tl4+l = (tlt2t3 + t2t3t4 +...) + (t2 + t2t3 + * * *) t+
= the sum three at a time of the (n + 1) tangents
and so on.
Hence we see that the same rule holds for (n+ 1)
angles as for n angles.
Hence, if the theorem be true for n angles, it is true
for (n + 1) angles.
But, by Arts. 98 and 124, it is true for 2 and 3 angles.
Hence the theorem is true for 4 angles; hence for
5 angles.... Hence it is true universally.
Cor. If the angles be all equal and there be n of
them and each equal to 0, then
s5 = n. tan 0; s2 = 'C2 tan2 0; s3 = n3 tan3 0.......