1.

Evaluate `cosacos2acos3a cos 999 a ,`where `a=(2pi)/(1999)dot`

Answer» Let `P = cosacos2acos3a...cos999a`
Let `Q = sinasin2asin3a...sin999a`
Then, `2^999PQ = (2sinacosa)(2sin2acos2a)...(2sin999acos999a)`
`= sin2asin4a...sin1996asin1998a`
`=(sin2asin4a...sin998a)((-sin(2pi-1000a))(-sin(2pi-1002a))...(-sin(2pi-1998a)))`...[As `-sin(2pi-theta) = sintheta`]
`=(sin2asin4a...sin998a)((-sin(2pi-1000((2pi)/1999)))(-sin(2pi-1002((2pi)/1999))))...(-sin(2pi-1998((2pi)/1999)))))`...[As `a = (2pi)/1999`]
`=(sin2asin4a...sin998a)((-sin((2pi)/1999(999)))(-sin((2pi)/1999(997)))...(-sin((2pi)/1999)))`
`=(sin2asin4a...sin998a)(sin999asin997a...sin3asina)`
`=>2^999PQ = sinasin2asin3a...sin999a`
`=>2^999PQ = = Q`
`=>P = 1/(2^(999))`
`:. cosacos2acos3a...cos999a = 1/(2^(999))`


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