A curve \(C\) has parametric equations \[x=\sec\theta+1,\qquad y=\tan\theta-2,\qquad -\frac\pi2<\theta<\frac\pi2.\]
Find the equations of the tangents to \(C\) at \(x=3\). Hence find the acute angle between the two tangents.[5]
Find the cartesian equation of \(C\). You need not state any restrictions on the values of \(x\).[1]
Using your answer to part (b), find the volume generated when the region enclosed by \(C\) and the line \(x=3\) is rotated \(2\pi\) radians about the \(x\)-axis.[3]