Numerator

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N = sin(n+1)A - sin(n-1)A

= 2cos[{(n+1)A+(n-1)A}/2]sin[{(n+1)A-(n-1)A}/2]

= 2cos(2nA/2)sin(2A/2)

= 2cos(nA)sin(A)

Denominator

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D = cos(n+1)A+2cos(nA)+cos(n-1)A

= 2cos(nA)+cos(n+1)A+cos(n-1)A

= 2cos(nA)+2cos[{(n+1)A+(n-1)A}/2]cos[{(n+1)A-(n-1)A}/2]

= 2cos(nA)+2cos(nA)cosA

So D=2cos(nA)(1+cosA)

So,

N/D

=2cos(nA)sin(A)/2cos(nA)(1+cosA)

= sinA/(1+cosA)

= 2sin(A/2)cos(A/2)/2cos^{2}(A/2)

=sin(A/2)/cos(A/2)

=tan(A/2)