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Question

Solve : cos12cos24cos36cos48cos60cos72cos96

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Solution

cos120.cos240.cos360.cos480.cos600.cos720.cos960=?
From trigonometry we have a following identity,
cosθ.cos(600θ).cos(600+θ)=14cos3θ
So, if θ=120
cos120.cos(600120).cos(600+120)=14cos(3×12)
cos120.cos480.cos720=14cos360
cos120.cos480.cos720=5+116(1)
if θ=240
cos240.cos(600240)cos(600+240)=14cos(3×240)
cos240.cos360.cos840=14cos720
cos240.cos360.cos840=5116(2)
Multiplying (1) & (2) we get
cos120.cos240.cos360.cos480.cos720.cos840=(5+116)(5116)(3)
We are asked to find the value of :
cos120.cos240.cos360.cos480.cos600.cos720.cos960
=cos120.cos240.cos360.cos480.cos720.cos(1800840).cos600
=cos120.cos240.cos360.cos480.cos720.(cos840).cos600
=(5+1)(15)16×16×12 (from (3) & cos600=1/2)
=1532×16
=432×16
=1128

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