= Solution
At their intersection, $x^2+y^2=1$ and $z^2=1$. The <gradients> of the defining functions give <normal vectors> to the <sphere> and <cylinder>:
$$
\mathbf n_s=(2x,2y,2z),\qquad \mathbf n_c=(2x,2y,0).
$$
The acute angle between the <tangent planes> is the acute angle between these <normal vectors>. Their <dot product> gives
$$
\cos\alpha=\frac{|\mathbf n_s\cdot\mathbf n_c|}{|\mathbf n_s|\,|\mathbf n_c|}
=\frac4{(2\sqrt2)(2)}=\frac1{\sqrt2},\qquad
\boxed{\alpha=\frac\pi4=45^\circ.}
$$
It is the same at every intersection point; the supplementary oriented angle is $135^\circ$.
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