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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 164Let a=sinπ14sin3π14sin5π14sin7π14sin9π14sin11π14sin13π14⇒a=sinπ14sin3π14sin5π14sinπ2sin(π−5π14)sin(π−3π14)sin(π−π14)⇒a=(sinπ14sin3π14sin5π14)2 [∵sin(π−x)=sinx]a=⎛⎜ ⎜ ⎜ ⎜⎝(2sinπ14cosπ14)(2sin3π14cos3π14)(2sin5π14cos5π14)8cosπ14cos3π14cos5π14⎞⎟ ⎟ ⎟ ⎟⎠2⇒a=⎛⎜ ⎜ ⎜ ⎜⎝sinπ7sin3π7sin5π78sin(π2−π14)sin(π2+3π14)sin(π2−5π14)⎞⎟ ⎟ ⎟ ⎟⎠2 {∵2sinθcosθ=sin2θ}⇒a=⎛⎜ ⎜ ⎜⎝sinπ7sin3π7sin5π78sin3π7sin5π7sinπ7⎞⎟ ⎟ ⎟⎠2=(18)2=164

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