( )

cos a cos 2a cos 3a..........cos 999a, a =

Let P denotes the desired product, and let

Q = sin a sin 2a sin 3a.......sin 999a.

Then

2 999 PQ = (2 sin a cos a)(2 sin 2a cos 2a).........(2 sin 999a cos 999a)

= sin 2a sin 4a .....sin 1998a

= (sin 2a sin 4a ........sin 998a) [–sin(2 π – 1000a)]. [–sin(2 π – 1002a)]........[–sin(2 π – 1998a)]

= sin 2a sin 4a ........sin 998a sin 999a sin 997a .......sin a = Q

It is easy to see that Q ≠ 0. Hence the desired product is P = .

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