Plane trigonometry, by S.L. Loney.

[Exs. XLIV.] SIMPLE TRIGONOMETRICAL SERIES.

287

24. ABCD... is a regular polygon of n sides which is inscribed in a
circle, whose centre is O and whose radius is r, and P is any point on the
arc AB such that POA is 0. Prove that
PA. PB +PA. PPC +PA. PD+PB. PC...
=r2 [2 cos2 ( - )cosec2 - n.
25. Two regular polygons, each of n sides, are circumscribed to and
inscribed in a given circle. If an angular point of one of them be joined
to each of the angular points of the other then the sum of the squares of
the straight lines so drawn is to the sum of the areas of the polygons as
n
26. A1, A2...A2n+l are the angular points of a regular polygon inscribed in a circle and 0 is any point on the circumference between A1
and A2n,+; prove that
OA +OA A +... + OA2n+l= OA2 + OA4 +... + OA2,,
27. If perpendiculars be drawn on the sides of a regular polygon of n
sides from any point on the inscribed circle whose radius is a, prove that
22 and
n \ n a}