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Prove that: ((sin7x+sin5x)+(sin9x+sin3x)...

Prove that: `((sin7x+sin5x)+(sin9x+sin3x))/((cos7x+cos5x)+(cos9x+cos3x))=tan6x`

Text Solution

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LHS
`((sin7x+sin5x)+(sin9x+sin3x))/((cos7x+cos5x)+(cos9x+cos3x)`
`(2sin(7x+5x)/(2)cos(7x-5x)/(2)+2sin(9x+3x)/(2)cos(9x-3x)/(2))/(2cos(7x+5x)/(2)cos(7x-5x)/(2)+2cos(9x+3x)/(2)cos(9x-3x)/(2))`
`=(2sin6xcosx+2sin6xcos3x)/(2cos6xcosx+2cos6xcos3x)`
`(2sin6xcosx+2sin6xcos3x)/(2cos6xcosx+2cos6xcos3x)`
`=(sin6x)/(cos6x)=tan6x` = RHS Hence Proved.
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