Plane trigonometry, by S.L. Loney.

[Misc. Exs. LXIX.] EXAMPLES. 479
15. ABCD... is a regular polygon of n sides which is inscribed in a
circle, whose radius is a and whose centre is O; prove that the product of
the distances of its angular points from a straight line at right angles to
OA and at a distance b (> a) from the centre is
bn [COSn ( sin- - sin (1 sin-'l b '
16. Prove that there is one, and only one, solution of the equation
= cos and that it is less than 4
4'
17. Prove that the general value of 6 which satisfies the equation
(cos 0 + i sin 0) (cos 20 + i sin 20)...... to n factors= 1
417r
its -+- (n, where n is any integer.
18. Prove that
e7r+e-7=2{l+22} 41+(2 ) 4 1+(.j......adinf.
19. Prove that
x3 x6 X9,3x, I - 3
1+ +..ad inf.=- x e 2 cos 2
1+13 + [ L E+ 3
20. Prove that
27r 47r 67r 147r 167r 1
cos -7 + cos- + cos 7 +.5. + cos + cs -- -2'
27r 47r 147r 167
and see - + sec +... + sec- + sec =8.
~~17 17 17 ~17
21. From the sum of the series
1 1
sin 20 - sin 40 + sin 60 -... ad inf.,
2 3
or otherwise, shew that
rr2 1 1 1 1 1
1-+ - - - + + - ad inf.
4 3 5 7 911
22. Prove that
1 sinhx 1 o s 4 1 1
2x cosh x - cos a a2 +x2 7 n (2n7r - a)2 + x2 (2n7r + a)2 + 2f