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If `x=sin(theta+(7pi)/(12))+sin(theta+pi/(12))+sin(theta+(3pi)/(12))dot``Y=cos(theta+(7pi)/(12))+cos(theta+pi/(12))+cos(theta+(3pi)/(12))dot`then prove that `X/Y-Y/X=2tan2theta`

Answer» `X=2sin(theta+pi/4)cos(pi/3)+sin(theta+3/12pi)`
`X=2sin(theta+pi/4)`
`Y=2cos(theta+pi/4)cos(pi/3)+cos(theta+pi/4)`
`Y=2cos(theta+pi/4)`
`X/Y-Y/X=(X^2-Y^2)/(XY)`
`=(4sin^2(theta+pi/4)-4cos^2(theta+pi/4))/(4sin(theta+pi/4)cos(theta+pi/4))`
`=(-4(cos^2(theta+pi/4)-sin^2(theta+pi/4)))/(2(2sin(theta+pi/4)cos(theta+pi/4))`
`=(-2cos2(theta+pi/4))/(sin2(theta+pi/4))`
`=-2(-sin2theta)/(cos2theta)`
`=(2sin2theta)/(cos2theta)`
`=tan2theta`.


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