1.

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

Answer» Here, we will use,
`sinA+sinB = 2sin((A+B)/2)cos((A-B)/2)`
`cosA+cosB = 2cos((A+B)/2)cos((A-B)/2)`
`L.H.S. = ((sin7x+sin5x)+(sin9x+sin3x))/((cos7x+cos5x)+(cos9x+cos3x))`
`=(2sin6xcosx+2sin6xcos3x)/(2cos6xcosx+2cos6xcos3x)`
`=(2sin6x(cosx+cos3x))/(2cos6x(cosx+cos3x))`
`=(sin6x)/(cos6x) = tan6x = R.H.S.`


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