Parameter Elimination with Products of Sines and Cosines in N-dimensions

Question

Given N coordinate dimensions (the $x$ 's) and M parameters (the $u$ 's) with M = N - 1

$x^1 = a ~cos~u^1$

$x^2 = a ~sin~u^1~cos~u^2$

...

$x^{N-1} = a ~sin~u^1~sin~u^2~sin~u^3 ... sin~u^{N-2} ~cos~u^{N-1}$

$x^N = a ~sin~u^1~sin~u^2~sin~u^3 ... sin~u^{N-2} ~sin~u^{N-1}$

We want to do parameter elimination to come up with the $F$ such that

$F(x^1,x^2,...,x^N) = 0$.

Can someone show me how to do this? This is just for my interest in learning, not for a homework. It comes from the first chapter of Synge and Schild Tensor Calculus (1949 - a Dover reprint). Thank you.

Answer 1

$Σ^N_{i=1} (x^i)^2 ~−~a^2 = 0$

Using the relation $sin^2x+cos^2x = 1$, and a bunch of factoring and simplifying. Easier to see by starting with the higher order terms rather than the lower order ones. Solved with the comment/hint of @player3236.