The classic infinite product representation of Cos(x) is:
Cos(x) = Π (n = 1 to ∞) ( 1 - 4x² / π²(2n-1)² )
From this, can you derive:
Cos(x) = Π (n = 1 to ∞) ( Π (n = 1 to ∞) ( (π²n² - 4x²) / (π²n² - x²) )
Derive the infinite product representation of Cos(x) from?
2012-02-25 5:44 am
回答 (1)
2012-02-25 8:56 am
✔ 最佳答案
Using solely the second relation,sin x = x Π ( 1 - x^2 / (πn)^2 ) = x Π ( (πn)^2 - x^2) ) / (πn)^2
sin 2x = 2x Π ( 1 - 4x^2 / (πn)^2 ) = 2x Π ( (πn)^2 - 4x^2) ) / (πn)^2
Taking the ratio of the above two equations, we get
Π Π ( (πn)^2 - 4x^2) ) / ( (πm)^2 - x^2) ) = (sin(2x)/2) / sin x = (sin x cos x) / sin x = cos x.
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