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Show that sin x+sin3x+sin5x+sin7x=4 sin4...

Show that `sin x+sin3x+sin5x+sin7x=4 sin4xcos2x cos x`.

Text Solution

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LHS `=sinx+sin3x+sin5x+sin7x = (sin7x+sin5x)+(sin5x+sin3x)`
`=2sin(7x+x)/(2)cos(7x-x)/(2) + 2sin(5x+3x)/(2)cos(5x-3x)/(2)`
`=2sin4xcos3x+2sin4xcosx`
`=2sin4x(cos3x+cosx)`
`=2sin4x[2cos(3x+x)(2)cos(3x-x)/(2)]`
`=2sin4x[2cos2xcosx]`
`=4cosxcos2xsin4x`= RHS Hence Proved.
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