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Question

State true or false:
A1,A2,A3...An are the vertices of a regular polygon of n sides. |OA1|=1. Then,
|A1A2||A1A3|....|A1An|=n.

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Solution

From part (i), |A1Ar|=|z1|1e2(r1)πi/n
=1e2(r1)πi/n[|z1|=1]
Hence |A1A2|.|A1A3|......|A1An|
=1e2πi/n1e4πi/n.......1e2(n1)πi/n ...(1)
Since e2πi/n,e4πi/n,.....e2(n1)πi/n are the n-1 imaginary, nth roots of unity, we have the identity
zn1(z1)(ze2πi/n)(ze4πi/n)......(ze2(n1)πi/n)
or zn1z1(ze2πi/n)(ze4πi/n)......(ze2(n1)πi/n)
or 1+z+z2+....+zn1(ze2πi/n)......(ze2(n1πi/n))
Putting z=1 in the above identity, we get
n=(1e2πi/n)(1e4πi/n).....(1e2(n1)πi/n)
Hence n=|n|=1e2πi/n1e4πi/n......1e2(n1)πi/n ...(2)
From (1) and (2), we get
|A1A2|.|A2A3||A1An|=n.

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