Plane trigonometry, by S.L. Loney.

436 TRIGONOMETRY. [Exs. LXIV.]
7r 37r 57r 2n- 1 n1r
22. 2-1 cos - cos - cos -...... cos 2n 1 =cos2n 2n 2n 2n 2. 37r. 57r 2n-1
23. 2"-1 sin -sin - sm -...... sin 2 nr=1.
22n 2n 2n 2n
7r 27r (2n-1)r (-1)n-1
24. cos cos......c o s = 2-1
n n n 22fl-1
25. Prove that
xn - acosn 1 rn- \ n )
x2 - 2anx cos nO + a2n nxn-"1 r- X2a os ( 2r7r a
x2- 2ax cos 0 + -) + a2
n
[In the expression (3) of Art. (362) change x into x + h, expand and
equate coefficients of h.]
26. The circumference of a circle of radius r is divided into 2n equal
parts at points P1, P2,...... P2; if chords be drawn from P1 to the other
points, prove that
P1P2. P1P3......P1Pa =?>l.
Also, if 0 be the middle point of the arc PP2n, prove that
OP1. OP2...... P,n= /2 rn.
27. If AxA2......A 2n be a regular polygon of n sides, inscribed in a
circle of radius a, and OA.+, be a diameter, prove that
OA1. OA2...... OAn = an.
28. A1A2......A is a regular polygon of n sides. From O the centre
of the polygon a line is drawn meeting the incircle in P1 and the circumcircle in P2.
Prove that the product of the perpendiculars on the sides drawn from
P1 is to the product of the perpendiculars from P2 as
w nO
cos - cot2 to 1,
n 2
0 being the angle between OPP1 and OA1.
29. ABCD...... is a regular polygon which is inscribed in a circle of
radius a and centre 0; prove that
PA2. PB2. PC2...... = r2 - 2ar" cos nO + a2n,
where OP is r and the angle AOP is 0.
Prove also that the sum of the angles that AP, BP, CP....... make
r~ sin n6
with OP is tan-l r" sin nO
rn cos nO - an