sin(A+B) = sinAcosB + cosAsinB (1)
sin(A-B) = sinAcosB - cosAsinB (2)
(1)-(2)
2cosAsinB = sin(+B) -sin(A-B)
2sin(1/2)θ (1+cosθ + cos2θ +...+ cosnθ )
=2sin(1/2)θ + 2cosθsin(1/2)θ+ ....+ 2cosnθsin(1/2)θ
=2sin(1/2)θ + [sin(3/2)θ - sin(1/2)θ]+ ....+ [ sin(n+(1/2)θ) -sin(n-(1/2)θ) ]
=2sin(1/2)θ +sin(n+(1/2)θ) - sin(1/2)θ
=sin(1/2)θ +sin(n+(1/2)θ)
=>
(1+cosθ + cos2θ +...+ cosnθ ) = 1/2 + sin(n+(1/2)θ)/ sin(1/2)θ
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