Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 2A / iii / Solution

Sinceintegration givesPart (i) gives $D=0$, so the tangent addition functional equation has the solutionsfor constants $C$ such that no pole lies in $(-a,a)$. Conversely, the tangent addition formula verifies the functional equation. If $f^{\prime }(0)=2$, then $C=2$, andon any such interval, necessarily with $a\le \pi /4$.