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Prove that sin x + sin 3x + sin 5x + si...

Prove that sin x + sin 3x + sin 5x + sin 7x=4 sin 4x cos 2x cos x.

Text Solution

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We have
`LHS =(sin (7x+ sin x) = (sin 5x + sin 3x)`
`=2 " sin " ((7x+x)/(2)) " cos " ((7x-x)/(2)) + 2 " sin " ((5x+ 3x)/(2)) " cos " ((5x -3x)/(2))`
`=2 " sin " 4x "cos " 3x + 2 " sin " 4x "cos " x`
` =2 " sin ' 4x (cos " 3x + " cos " x)`
`=(2 " sin " 4x) xx 2 " cos " ((3x +x)/(2)) " cos " ((3x -x)/(2))`
` =" (2 ' sin4x) xx 2 cos 2x cos x )"`
` =4 " sin 4x cos 2x cos x = RHS"`
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