Plane trigonometry, by S.L. Loney.

448

TRIGONOMETRY.

EXAMPLES. LXV.
Prove that
1 1 1 1 r2
1. - + - +...... ad inf. =
2. 1 1 _ _ _ _
2. - i6 + + 8...... ad inf= 6
ad inf6 (=6r
1 1 1 1 7r2
3. 72+ 24+36+48+...... ad inf.=yi
1 3 6 10 7r2 /.r2\
4. + +t + -+..... ad inf. =64 1-
5. Prove that the sum of the products, taken two and two together,, r4
of the reciprocals of the squares of all odd numbers is 3-84
6. Prove that the sum of the products, taken two and two together,
7r4
of the reciprocals of the squares of all even numbers is 12-0
Prove that
1 20 20
7. cot a0- _. 0 - 222 _ 02- ''
- ^ 1. 1 i 1 + 12 adinf.
' 1 1 1 1 1
$ —$- + +...... ad inf.
G — 7r + + 2+d
8. cosec0= - r 0 + 0-27r 0+27r 0-37r 0+3
_ 1 _ _ _ _ _ _
+ 20 2 02 _n22
n==1 /
and hence that
1 + 0 cosec 0 1 1
262 02 02W2+ 02_ 222-...... ad inf.
[Use the relation cosec 0= (tan +cott).]
91 1 _ec 0= ~ 3 + ___ ~...... ad inf.
947r sec = r _ 402 32r2 _ 402 52r2 2 402.
[Use the relation 2 sec O= tan +) + cot( +).]