Prove that cotπ16⋅cot2π16⋅cot3π16…cot7π16=1

cotπ16⋅cot7π16⋅cot2π16⋅cot6π16⋅cot3π16⋅cot5π16⋅cot4π16cotπ16⋅cotπ2−π16cot2π16⋅cotπ2−2π16cot3π16⋅cotπ2−3π16⋅cot4π16=cotπ16⋅tanπ16cot2π16⋅tan2π16cot3π16⋅tan3π16⋅cotπ4=1×1×1×1 (∵cotθ tanθ=1)=1

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Prove that $\cot \frac{π}{16} \cdot co t \frac{2 π}{16} \cdot \cot \frac{3 π}{16} \dots \cot \frac{7 π}{16} = 1$

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Detailed Solution

$(\cot \frac{π}{16} \cdot \cot \frac{7 π}{16}) \cdot (\cot \frac{2 π}{16} \cdot \cot \frac{6 π}{16}) \cdot (\cot \frac{3 π}{16} \cdot \cot \frac{5 π}{16}) \cdot \cot \frac{4 π}{16}$

$(\cot \frac{π}{16} \cdot \cot (\frac{π}{2} - \frac{π}{16})) (\cot \frac{2 π}{16} \cdot \cot (\frac{π}{2} - \frac{2 π}{16}))$ $(\cot \frac{3 π}{16} \cdot \cot (\frac{π}{2} - \frac{3 π}{16})) \cdot \cot \frac{4 π}{16}$

$= (\cot \frac{π}{16} \cdot \tan \frac{π}{16}) (\cot \frac{2 π}{16} \cdot \tan \frac{2 π}{16}) (\cot \frac{3 π}{16} \cdot \tan \frac{3 π}{16}) \cdot \cot \frac{π}{4}$