Find a vector parallel to the line of intersection of the tangent planes...?

No, you need to think "out of the box"....consider that the 1 sphere is the level surface of the function F[x,y,z] = x^2 + y^2 + z^2, and you should know that grad F is a vector which is perpendicular to the surface...normal to the tangent plane. Grad F = < 2x , 2y, 2z >. With the rho, theta , phi given you can find the two vectors and then a cross operation will yield the vector parallel to the line of intersection.