Theorem 5.4c: The area of the region in the figure above is

2r2cos2(180k/n)sin2(180/n)sec(180j/n)tan(180(j-1)/n)sec(180(j-2)/n)

proof2: From Corollary 5.1b we know that the area is sjsj-1sin(360(j-1)/n). As in the previous proof,

sjsj-1sin(360(j-1)/n) =

 

2r2cos2(180k/n)sin2(180/n)sec(180j/n)sec2(180(j-1)/n) sin(180(j-1)/n)cos(180(j-1)/n)sec(180(j-2)/n)

 

= 2 r2cos2(180k/n)sin2(180/n) tan(180(j-1)/n)sec(180j/n)sec(180(j-2)/n)

proof 1

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