Generalizing a Trigonometric Infinite Product of Vieta
Solution 1:
Write $\ a_n(x) := \sin(nx)/\sin(x) \ $ in terms of $\,\cos(kx).\,$ The first examples are: $$ a_2(x) = 2\cos(x), $$ $$ a_3(x) = 1 + 2\cos(2x), $$ $$ a_4(x) = 2\cos(x) + 2\cos(3x), $$ $$ a_5(x) = 1 + 2\cos(2x) + 2\cos(4x), $$ $$ a_6(x) = 2\cos(x) + 2\cos(3x) + 2\cos(5x). $$ The pattern is now obvious. Thus, the general infinite product with $\,n>1\,$ is: $$ \frac{\sin x}x = \prod_{k=1}^ \infty \frac1n a_n\Big(\frac{x}{n^k}\Big). $$