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Prove that: sinA.sin(60^(@)+A).sin(60^...

Prove that:
`sinA.sin(60^(@)+A).sin(60^(@)-A)=1/4sinA`

Text Solution

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LHS `=sinA.sin(60^(@)+A).sin(60^(@)-A)`
`=1/2sinA.[2sin(60^(@)+A)-(60^(@)-A)]`
`1/2sinA[cos{(60^(@)+A)-(60^(@)-A)}]-cos{(60^(@)+A)+(60^(@)-A)}`
`=1/2sinA[cos2A-cos120^(@)]`
`=1/2[cos2AsinA-cos(90^(@)+30^(@)).sinA]`
`=1/4[2cos2AsinA+2sin30^(@)sinA]`
`=1/4[sin(2A+A)-sin(2A-A)=2.1/2.sinA]`
`=1/4[sin3A-sinA+sinA]`
`=14sin3A=R.H.S` Hence Proved.
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