Plane trigonometry, by S.L. Loney.

SUMMATION OF SERIES.

415

Let Ur- the rth term, i.e.
tan {a + (r - 1) } tan {a + r3,.. (u +l) tan p
=[1 + tan {a+ (r - 1) } tan {a +rf3}] x tan [a+r3 - (a + r - 1)]
=tan{a+r}) -tan{a+qr-1/}. [Art. 98.]
Hence giving r in succession the values 1, 2,......n, we have
(1 + ul) tan = tan (a + 3) - tan a,
(1 +u2) tan 3= tan (a + 2/) - tan (a +3 ),.................................
(1 + u,) tan/3= tan {a + n/3} - tan {a + (n - 1) }.
Hence by addition
(n + S,) tan = tan (a + nj) - tan a,
so that tSz tan (a + n/) -tan a- n tan
so that S== = - ^ --
tan /
EXAMPLES. LXII.
Sum the series
1. cosec 0 + cosec 20 + cosec 40 +...... to n terms.
2. cosec 0 cosec 28 + cosec 20 cosec 30 + cosec 30 cosec 4 +......to n
terms.
3. sec 8 sec 20 + sec 20 sec 30 + sec 30 sec 40 +...... to n terms.
4. sec 0 sec (8 + 0) + sec (0 + 0) sec ( - + 2b) + sec (8 + 20) sec (O + 30)
+...... to n terms.
1 1 1
5. -.+ + +... to n terms.
cos acos3a + cosa + cos 5a cos a + cos 7a
1 8 1 0 11 8
6. tan 0 + tan + 2 tan + tan +...... ad inf.
1 8 1 8 1 8
7. tanh 0 + tanh + tanh + I tanh +...... to n terms.
2 2 2!2 22 2:3 2"
8. tan 0 sec 20 + tan 20 sec 40 + tan 40 sec 80 +...... to n terms.
0 0 0 0 0
9. tan sec + tan s a see + tan -3 sec -+...... to n terms and to
infinity.
1 1 1
10. os + +...... to n terms.
2 cos 0 22 cos 0 cos 20 23 cos 0 cos 20 cos 22 0