1.

Prove that `4cos12^@cos48^@cos72^@=cos36^@`

Answer» Here, we will use the following rules,
`2cosAcosB = cos(A+B)+cos(A-B)`
`cosA+cos B = 2cos((A+B)/2)cos((A-B)/2)`
`L.H.S. = 4cos12^@cos48^@cos72^@`
`=2cos72^@(cos(48+12)^@+cos(48-12)^@)`
`=2cos72^@(cos60^@+cos36^@)`
`=2cos72^@(1/2+cos36^@)`
`=cos72^@ +2cos72^@cos36^@`
`=cos72^@ +cos(72+36)^@+cos(72-36)^@`
`=cos72^@+cos108^@+cos36^@`
`=2cos((108+72)/2)^@cos((108-72)/2)^@+cos36^@`
`=2cos90^@cos36^@+cos36^@`
`=0+cos36^@ = cos36^@=R.H.S.`


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