Past exam of the mathematical methods II course of the University of Oxford / 2021 / 4 / 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:The acute angle between the tangent planes is the acute angle between these normal vectors. Their dot product givesIt is the same at every intersection point; the supplementary oriented angle is $135^{\circ }$.