UPSC CDS 2 2022 Math Paper

Concept:In a regular polygon, the inscribed circle touches each side at its midpoint.The radius is perpendicular to the side, and the angle at the centre subtended by one side is $\frac{360^{\circ}}{n}$.Use the tangent ratio in a right triangle formed by the centre, the midpoint of a side, and a vertex.Explanation:For a regular polygon with $15$ sides, the central angle is $\frac{360^{\circ}}{15} = 24^{\circ}$.In the right triangle $\text{triangle OAC}$, where $O$ is the centre, $C$ is the midpoint of a side, and $A$ is one end of that side:• $OC$ is the radius $r$ (perpendicular to side $AB$).• $AC$ is half the side length: $AC = \frac{1}{2}$ (since side $= 1$ unit).• $\text{angle AOC}$ is half the central angle: $\text{angle AOC} = \frac{24^{\circ}}{2} = 12^{\circ}$.From $\text{triangle OAC}$: $\tan 12^{\circ} = \frac{AC}{OC} = \frac{\frac{1}{2}}{r}$.Thus $r = \frac{\frac{1}{2}}{\tan 12^{\circ}} = \frac{1}{2} \cot 12^{\circ}$.Diameter $= 2r = 2 \times \frac{1}{2} \cot 12^{\circ} = \cot 12^{\circ}$.Answer:$\cot 12^{\circ}$ (Option B).