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If A1,A2,...An are vertices of regulr polygen of n sides, inscribed in circle of radius r, whose cnetre is origin O, any P is any point on the arc An,A1 such that POA1=θ. The value of sum of lenghts of the lines joining P to the angluar points of the polygen is

A
2r.sec π2n.sin (θ2+π2n)
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B
2r.cosecπ2n.cos(θ2+π2n)
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C
2r.cosecπ2n.cos(θ2π2n)
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D
2r.sec π2n.sin (θ2π2n)
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Solution

The correct option is C 2r.cosecπ2n.cos(θ2π2n)
According to problem,

1OA2=A2OA3=...AnOA0=2πn


POA1=θ (Given)
POA2=θ+2πn
POA3=θ+2πn...
Hence if r be the radius of the circle, we have
PA1=2r sinPOA12=2r sinθ2
PA2=2r sinPOA22=2r sin(θ2+πn)
Hence the required sum
=2r[sinθ2+sin(θ2+πn)+sin(θ2+2πn)+...n terms]
=2r[sinθ2+(n12)πnsinnπ2n]sinπ2n
=2r cosec π2nsin[π2+θ2π2n]
=2r cosec π2n cos(θ2π2n)

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