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Question

Find the value of sinπ14.sin3π14.sin5π14.sin7π14sin9π14sin11π14.sin13π14.

A
164
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B
364
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C
564
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D
764
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Solution

The correct option is C 164
Let a=sinπ14sin3π14sin5π14sin7π14sin9π14sin11π14sin13π14
a=sinπ14sin3π14sin5π14sinπ2sin(π5π14)sin(π3π14)sin(ππ14)
a=(sinπ14sin3π14sin5π14)2 [sin(πx)=sinx]
a=⎜ ⎜ ⎜ ⎜(2sinπ14cosπ14)(2sin3π14cos3π14)(2sin5π14cos5π14)8cosπ14cos3π14cos5π14⎟ ⎟ ⎟ ⎟2
a=⎜ ⎜ ⎜ ⎜sinπ7sin3π7sin5π78sin(π2π14)sin(π2+3π14)sin(π25π14)⎟ ⎟ ⎟ ⎟2 {2sinθcosθ=sin2θ}
a=⎜ ⎜ ⎜sinπ7sin3π7sin5π78sin3π7sin5π7sinπ7⎟ ⎟ ⎟2=(18)2=164

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