Question

The value of (cot35^(@)-1)(cot32^(@)-1)(cot29^(@)-1)*(cot26^(@)-1) (cot23^(@)-1)*(cot22^(@)-1)*(cot19^(@)-1)*(cot16^(@)-1) *(cot13^(@)-1)*(cot10^(@)-1) is equal to​

Answer 1

$\textbf{Given:}$

$\mathsf{(cot35^\circ-1){\times}(cot32^\circ-1){\times}(cot29^\circ-1){\times}(cot26^\circ-1){\times}(cot23^\circ-1){\times}}$

$\mathsf{(cot22^\circ-1){\times}(cot19^\circ-1){\times}(cot16^\circ-1){\times}(cot13^\circ-1){\times}(cot10^\circ-1)}$

$\textbf{To find:}$

$\textsf{The value of}$

$\mathsf{(cot35^\circ-1){\times}(cot32^\circ-1){\times}(cot29^\circ-1){\times}(cot26^\circ-1){\times}(cot23^\circ-1){\times}}$

$\mathsf{(cot22^\circ-1){\times}(cot19^\circ-1){\times}(cot16^\circ-1){\times}(cot13^\circ-1){\times}(cot10^\circ-1)}$

$\textbf{Solution:}$

$\textbf{Formula used:}$

$\boxed{\begin{minipage}{6.5cm} \\\\\mathsf{If\;A\;\&\;B\;are\;acute\;angles\;with\;A+B=45^\circ}\\\\\mathsf{then\;(cotA-1)(cotB-1)=2}\\ \end{minipage}}$

$\mathsf{Consider,}$

$\mathsf{(cot35^\circ-1){\times}(cot32^\circ-1){\times}(cot29^\circ-1){\times}(cot26^\circ-1){\times}(cot23^\circ-1){\times}}$

$\mathsf{(cot22^\circ-1){\times}(cot19^\circ-1){\times}(cot16^\circ-1){\times}(cot13^\circ-1){\times}(cot10^\circ-1)}$

$\mathsf{Rearranging}$

$\mathsf{=(cot35^\circ-1){\times}(cot10^\circ-1){\times}(cot32^\circ-1){\times}(cot13^\circ-1){\times}(cot29^\circ-1){\times}}$

$\mathsf{(cot16^\circ-1){\times}(cot26^\circ-1){\times}(cot19^\circ-1){\times}(cot23^\circ-1){\times}(cot22^\circ-1)}$

$\mathsf{=[(cot35^\circ-1){\times}(cot10^\circ-1)]{\times}[(cot32^\circ-1){\times}(cot13^\circ-1)]{\times}[(cot29^\circ-1){\times}}$

$\mathsf{(cot16^\circ-1)]{\times}[(cot26^\circ-1){\times}(cot19^\circ-1)]{\times}[(cot23^\circ-1){\times}(cot22^\circ-1)]}$

$\mathsf{Using\;the\given\;formula}$

$\mathsf{=2{\times}2{\times}2{\times}2{\times}2}$

$\mathsf{=32}$

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