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Can we express the distribution of a coordinate of the $n$-sphere in any known distribution?

In formal terms, consider $S^n = {xinmathbb{R}^{n + 1}: |x|=1}$ (i.e. the usual $n$-sphere). If we sample $x$ uniformly from $S^n$ what is the distribution of $x_1$?

By "sampling uniformly" I mean that any point in $S^n$ has the same value for the density probability function. And $x_1$ means the first coordinate of vector $x$.

$n=1$

(the circle)

$x_1$ follows the arcsine distribution.

$n=2$

(the sphere)

Thanks to Archimedes we know that $x_1$ follows the Uniform distribution.

$n>2$

Do we know?

...

I know that this is equivalent to ask the distribution of the dot product of two random points on the $n$-sphere. But I also do not know that!

$defd{mathrm{d}}$This is a direct application of the formula for the surface area of the hyperspherical cap. Denoting by $I(x; a, b)$ the regularized beta function, the surface area of an $(n + 1)$-dimensional hyperspherical cap with height $x leqslant 1$ and radius $1$ is$$
A_n(x) = frac{1}{2} A_n(2) Ileft( x(2 - x); frac{n}{2}, frac{1}{2} right),
$$
thus for $-1 leqslant x leqslant 0$,$$
P(X_1 leqslant x) = frac{A_n(x + 1)}{A_n(2)} = frac{1}{2} Ileft( 1 - x^2; frac{n}{2}, frac{1}{2} right).
$$

Since $dfrac{∂I}{∂x}(x; a, b) = dfrac{1}{B(a, b)} x^{a - 1} (1 - x)^{b - 1}$, then for $-1 < x < 0$,$$
f_{X_1}(x) = frac{d}{d x} P(X_1 leqslant x) = frac{(1 - x^2)^{frac{n}{2} - 1}}{Bleft( dfrac{n}{2}, dfrac{1}{2} right)}.
$$
By symmetry,$$
f_{X_1}(x) = frac{(1 - x^2)^{frac{n}{2} - 1}}{Bleft( dfrac{n}{2}, dfrac{1}{2} right)}. quad forall -1 < x < 1
$$
Indeed, for $n = 2$ this is a uniform distribution.

I came across this answer previously. The original question was about the marginals of a point on the unit sphere in $mathbb{R}^3$, but the poster added the case for the sphere in $mathbb{R}^d$ (see Added section).

His method is summarized as follows (I have replaced $d$ in his answer with $n+1$):

1) Let $Z$ be a multivariate Gaussian r.v. with $(n+1)$ components, then $ frac{Z}{lVert Z rVert}$ is uniformly distributed on the unit sphere $S^n$. Thus, we set $ X= frac{Z}{lVert Z rVert}$.

2) Rewrite the CDF of $X_1 = frac{Z_1}{lVert Z rVert}$ in terms of $frac{Z_1^2}{Z_2^2 + dots Z_n^2}$.

3) The ratio $frac{ncdot Z_1^2}{Z_2^2 + dots Z_n^2}$ follows an $F_{1,n}$ distribution. Thus the CDF of $ X_1 = frac{Z_1}{lVert Z rVert}$ can be determined from (2). Differentiating gives
$$f_{X_1}(x) = frac{1}{B bigl( frac{n}{2}, frac{1}{2} bigr)}(1-x^2)^{(n/2-1)}. $$