An inverse-tangent identity and an enclosed rational-curve area

1. By considering two perpendicular lines, explain why \[\tan^{-1}(3)-\tan^{-1}\left(-\frac13\right)=\frac{\pi}{2}.\][1]

The curves \(C_1\) and \(C_2\) have equations \[C_1:\ y=\frac{1}{1+x^2},\qquad C_2:\ y=\frac{k}{4x+3},\] where \(k>0\).

2. Find the set of values of \(k\) for which the curves have at least one point of intersection.[3]

For the remaining parts, take \(k=\frac32\).

3. Sketch both curves on the same axes, labelling their intersections with the coordinate axes and stating the equations of all asymptotes.[3]
4. Find the exact area of the finite region enclosed by the curves. Use the identity in part (i) to simplify your answer.[5]