Question

Prove that: `sin 3x + sin 2x + sin x = ``4 sin x cos``x/2``cos``(3x)/2`

Answer 1

Here, we will use,
`cosA+cosB = 2cos((A+B)/2)cos((A-B)/2)`
`sinA+sinB = 2sin((A+B)/2)cos((A-B)/2)`
`L.H.S. = sin3x+sin2x+sinx`
` = (sin3x+sinx)+sin2x`
`=2cos2xsinx+2sinxcosx`
`=2sinx(cos2x+cosx)`
`=2sinx(2cos((3x)/2)cos(x/2))`
`=4sinxcos((3x)/2)cos(x/2) = R.H.S.`