Chebyshev polynomials and Using these, you can express as a polynomial in , namely
The properties of Chebyshev polynomials that you need here are that, for n odd, is of the form a polynomial of degree n with leading coefficient and constant term 0. For n even, is of the form Again, the leading coefficient is and the constant term is 0.
For fixed x and n, the numbers are all solutions of the equation It follows that, if n is odd, then the numbers are all solutions of the equation But the product of the roots of an equation of odd degree is the negative of the constant term divided by the coefficient of the leading term. The product of the roots in this case is Notice that of those factors have a negative sign, so that product is equal to
The constant term in the equation is , and the coefficient of the leading term is Thus the formula for the product of the roots tells you that
In the case where n is even there is an extra complication caused by the term in the formula for This corresponds to the factor in the product given by . That factor is After stripping out that factor from both sides of the identity, you can use the same argument as for the case where n is odd.