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Question

# The value of sinπ14⋅sin3π14⋅sin5π14⋅sin7π14⋅sin9π14⋅sin11π14⋅sin13π14 is equal to

A
1
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B
116
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C
164
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D
None of these
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Solution

## The correct option is C 164Let I=sinπ14sin3π14sin5π14sin7π14sin9π14sin11π14sin13π14=sinπ14sin3π14sin5π14sinπ2sin(π−5π14)sin(π−3π14)sin(π−π14)=(sinπ14sin3π14sin5π14)Now,sinπ14=sin(π2−6π14)=cos6π14=cos(π−8π14)=cos8π14sin3π14=sin(π2−4π14)=cos4π14sin5π14=sin(π2−2π14)=cos2π14Let x=π14Then,I=(cos8x⋅cos4x⋅cos2x)2=((2sin2xcos2x)cos4xcos8x2sin2x)2=((2sin8xcos8x4sin2x)2=((2sin8xcos8x)8sin2x)2=(sin16x8sin2x)2=⎛⎜ ⎜ ⎜⎝sin16π148sin2π14⎞⎟ ⎟ ⎟⎠2=⎛⎜ ⎜ ⎜ ⎜⎝sin(π+2π14)8sin2π14⎞⎟ ⎟ ⎟ ⎟⎠2=⎛⎜ ⎜ ⎜⎝−sin2π148sin2π14⎞⎟ ⎟ ⎟⎠2=(−18)2=164

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