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Prove that: sin 12^0 sin 48^0 sin 54^0=1...

Prove that: `sin 12^0 sin 48^0 sin 54^0=1/8`

Text Solution

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LHS `=(1)/(2)[cos36^(@)-cos60^(@)]sin54^(@)=(1)/(2)[cos36^(@)sin54^(@)-(1)/(s)54^(@)]`
`=(1)/(4)[2cos 36^(@)sin54^(@)-sin54^(@)]=(1)/(4)[sin 90^(@)+sin18^(@)-sin54^(@)]`
`=(1)/(4)[1-2sin18^(@)cos36^(@)]`
`=(1)/(4)[1-(2 sin 18^(@))/(cos 18^(@))cos18^(@)cos36^(@)]=(1)/(4)[1-(sin36^(@)cos 36^(@))/(cos18^(@))]`
`=(1)/(4)p1-(2 sin 36^(@)cos36^(@))/(2 cos 18^(@))]=(1)/(4)[1-(sin72^(@))/(2sin72^(@))]=(1)/(2)[1-(1)/(2)]=(1)/(8)=RHS`.
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