i will call alpha as A
cotA1.cotA2.........cotAn=1
cosA1.cosA2,......cosAn=sinA1..........sinAn=X
X2=(cosA1.cosA2....cosAn)(sinA1.sinA2....sinAn)
= 1/2n(sin2A1)(sin2A2)....(sin2An)
X2≤1/2n since sin2A1≤1
X≤1/2n/2
hence maximum value of given expression is 1/2n/2