Question

If x=2pi/2009, then cosxcos2xcos3x....cos1004x = ?

Answer 1

@ganeshie8 @eliassaab

Answer 2

I did it numerically, I found the quantity equal
\[

5.8329*10^{-303}\approx 0
\]

Answer 3

Yes, you are going to have some values of cos@
very near zero, for example
in the vicinity of 250 x=(500/2009) pi = pi/4 approx.

Answer 4

But\[
\cos\left ( \frac \pi 4 \right)=\frac {\sqrt 2}{2}
\]

Answer 5

Yes, but multiplication by near-zero values wipes all else out.

Answer 6

You want to consider \[ 500 x \approx \frac \pi 2 \]

Answer 7

\[
\cos\left ( \frac \pi 2 \right)=0

\]

Answer 8

I am not getting equations from you but formatting instructions, hard to follow. As your numerical calculations confirmed, those near-zero cosines wipe out everything else.

Answer 9

Also all of theses cosines are < 1, the product of 1004 of them must be very small.