1.

Prove that: `sin12^(@)sin48^(@)sin54^(@)=1/8`

Answer» LHS `=sin12^(@)sin48^(@)sin54^(@)`
`=1/2.(2sin48^(@)sin12^(@)).sin(90^(@)-36^(@))`
`=1/2[cos(48^(@)-12^(@))-cos(48^(@)+12^(@))].cos36^(@)`
`=1/2[cos(48^(@)-12^(@))-cos(48^(@)+12^(@))].cos36^(@)`
`=1/2[cos36^(@)-cos60^(@)].cos36^(@)`
`=1/2[(sqrt(5)+1)/(4)-1/2].((sqrt(5)+1)/(3))`
`=1/2(sqrt(5)+1-2)/(4).(sqrt(5)+1)/(4)`
`=((sqrt(5)-1)(sqrt(5)+1))/(32)`
`=(5-1)/(32)=4/(32)=1/8` = RHS Hence Proved.


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