MISCELLANEOUS EXAMPLES ON CHAPTER III 
107 
47. Show that, if m and n are positive integers, and m S n, then 
x m ~ 1 /{\ +x n ) — {l/n) 2 a> m ~ n /{x — w), where w is a root of x n +1=0: and hence 
show that, if n is even, 
I 
l+.r n 
2 in-1 
- 2 
n o 
1 — X cos 
(2r + l) 
2x cos 
(2r + l) n 
n 
and find the corresponding formula when n is odd. 
48. Express x m ~ l /{l — x n ), where m and n are positive integers, and rn < n, 
in partial fractions, and obtain the formulae for 1/(1 — x n ) corresponding to 
those of Ex. 47. 
49. Show that 
x n — a n cos n6 
x 2n — 2x n a n cos ntì + a 2 
n-1 
2 ■ 
x-acos ( 6 + 
2rn 
r °.r 2 — 2,racos + +a ! 
50. If p 1 , pi, ■■■ p n tire the distances of a point P, in the plane of a 
regular polygon, from the vertices, prove that 
11 1 n r ‘2n _ a 2n 
2 —2 
i /V 
r 2 — a? r ln — 2r n a n cos nd + d ln ’ 
where 0 is the centre and a the radius of the circumcircle of the polygon, 
r the length OP, and 6 the angle between OP and the radius from 0 to any 
vertex of the polygon. 
51. If A x A<i... A n , BjB.2... B m are concentric regular polygons, m and n 
being prime to one another, prove that 
n m ^ wiTi h2mn __ ^2mn 
2 2 = 
r=i s=i (d r 5g) 2 h 2 —a 2 h 2mn - 2b mn a mn cos mn 6 + d lran ’ 
where a and b are the radii of the circumcircles of the polygons, and 6 the 
angle between any two radii drawn one to a vertex of each polygon. 
{Math. Trip. 1903.) 
52. If p and q are integers, and q prime to p, and k is an odd positive 
integer less than 2p, and 6 = qnjp, show that 
cos h (a-\-nd) 
2 —;——;—~—p cot pa, 
n=o sin(n + w(9) 
[We have pX “ 
Î ,-I sin k{a + nd) 
2 ; — p. 
n=o SHI (a + nd) 
}>—1 
X v —\ X— 1 ] x—t n ' 
where £=cos 26+isin 26, 1 <\^p. 
In this equation write £ (¿+1) for X and cos 2a-isin 2a for x.~\