Answer:
The value of given expression is \(\frac{1}{2^{999}}\).
Step-by-step explanation:
The given expression is
\(\cos A\cos 2A\cos 3A...........\cos 998A\cos 999A\)
where, \(A=\frac{2\pi}{1999}\)
Let as assume,
\(P=\cos A\cos 2A\cos 3A...........\cos 998A\cos 999A\)
\(Q=\sin A\sin 2A\sin 3A...........\sin 998A\sin 999A\)
\(2^{999}PQ=2^{999}(\cos A\cos 2A.........\cos 999A)\)\((\sin A\sin 2A........\sin 999A)\)
\(2^{999}PQ=(2\cos A\sin A)(2\cos 2A\sin 2A)...........(2\cos 998A\sin 998A)(2\cos 999A\sin 999A)\)
Using the formula, \(2\sin x\cos x=\sin 2x\), we get
\(2^{999}PQ=\sin 2A\sin 4A......\sin 1998A\)
\(2^{999}PQ=[\sin 2A\sin 4A......\sin 998A][-\sin(2\pi -1000A)][-\sin(2\pi -1002A)]...[-\sin(2\pi -1998A)]\) .... (1)
Now,
\(-\sin(2\pi -1000A)=-\sin(2\pi -1000(\frac{2\pi}{1999}))\)
\(-\sin(2\pi -1000A)=-\sin(\frac{2\pi\cdot 999}{1999})\)
\(-\sin(2\pi -1000A)=-\sin 999A\)
So, equation (1) can be written as
\(2^{999}PQ=[\sin 2A\sin 4A......\sin 998A][\sin 999A\sin 997...\sin A]\)
\(2^{999}PQ=\sin A\sin 2A\sin 3A...........\sin 998A\sin 999A\)
\(2^{999}PQ=Q\)
Divide both sides by Q.
\(2^{999}P=1\)
Divide both sides by \(2^{999}\)
\(P=\frac{1}{2^{999}}\)
Therefore the value of given expression is \(\frac{1}{2^{999}}\).
