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Question
Prove the following :
`(sinx - sin3x + sin5x - sin7x)/(cosx - cos3x - cos5x + cos7x)` = cot2x
Solution
L.H.S. = `(sinx - sin3x + sin5x - sin7x)/(cosx - cos3x - cos5x + cos7x)`
= `((sin5x + sinx) - (sin7x + sin3x))/((cosx - cos 5x) - (cos3x - cos7x)`
= `(2sin((5x + x)/2)*cos((5x - x)/2) - 2sin((7x + 3x)/2)*cos((7x - 3x)/2))/(2sin((x + 5x)/2)*sin((5x - x)/2) - 2sin((3x + 7x)/2)*sin((7x - 3x)/2)`
= `(2sin3x*cos2x - 2sin5x*cos2x)/(2sin3x*sin2x - 2sin5x*sin2x)`
= `(2cos2x(sin3x - sin5x))/(2sin2x(sin3x - sin5x))`
= `(cos2x)/(sin2x)`
= cot 2x
= R.H.S.
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