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cosx = 1-2[sin(x/2)]^2
1-cosx = 2[sin(x/2)]^2
(1/2)∫(π/2->3π/2) cosx/(1-cosx) dx
=(1/2)∫(π/2->3π/2) [1-2[sin(x/2)]^2]/[2[sin(x/2)]^2] dx
=(1/4)∫(π/2->3π/2) { [csc(x/2)]^2 - 2 } dx
=(1/4)[ -2cot(x/2) - 2x ]|(π/2->3π/2)
=(1/4)[ ( 0-3π) -(0-π) ]
=-π/2