JEE Trigonometry

Geometry Level 5

A regular nn sided polygon has vertices V1, V2,  , VnV_1, \ V_2, \ \cdots \ , \ V_n.

If V1V2+V1V3++V1V7cscπn=V1V2(1+cotπ242)\dfrac{\overline{V_1 V_2}+\overline{V_1 V_3}+ \cdots \cdots +\overline{V_1 V_7}}{\csc {\dfrac{\pi}{n}}}=\overline{V_1 V_2}\left(\dfrac{1+\cot {\dfrac{\pi}{24}}}{2}\right) then find the value of nn

Assumptions and Source\textbf{Assumptions and Source}

    \bullet \ \ \ \ V1Vi \overline{V_1 V_i} denotes the distance between the first vertex and the ithi^{\text{th}} vertex, where V1V_1 is adjacent to V2V_2, which is adjacent to V3V_3 and so on.

    \bullet \ \ \ \ A similar (but easier) question appeared in the 1994 IITJEE exam1994 \ \text{IITJEE exam}.

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