COMEDK 2025 Morning Solved Paper

Concept:The product $\tan k^\circ \tan (90^\circ - k)$ equals 1 for any $k$.This property helps simplify the given product to find the relation between $x$ and $y$.Explanation:The given condition is $\tan x^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 88^\circ \cdot \tan y^\circ = 1$.The angles from $2^\circ$ to $88^\circ$ can be paired as $(2^\circ, 88^\circ)$, $(3^\circ, 87^\circ)$, …, $(44^\circ, 46^\circ)$.For each pair, $\tan k^\circ \cdot \tan (90^\circ - k)^\circ = \tan k^\circ \cdot \cot k^\circ = 1$.The middle angle $45^\circ$ has $\tan 45^\circ = 1$.Thus the product $\prod_{k=2}^{88} \tan k^\circ = 1$.Substituting into the condition gives $\tan x^\circ \cdot 1 \cdot \tan y^\circ = 1$, so $\tan x^\circ \tan y^\circ = 1$.This implies $\tan y^\circ = \cot x^\circ = \tan (90^\circ - x^\circ)$, hence $y = 90 - x$ (considering principal values).Therefore $x + y = 90^\circ$.Then $\cot (x + y) = \cot 90^\circ = 0$.Answer:$0$