Question

A+B+C=360 cot A/4 + cot B/4 + cot C/4=k cot A/4 cot B/4 cot C/4 then find k

Answer 1

$\textbf{Given:}$

$A+B+C=360^{\circ}$ $\text{and}$

$\cot\frac{A}{4}+\cot\frac{B}{4}+\cot\frac{C}{4}=k\,\cot\frac{A}{4}\;\cot\frac{B}{4}\;\cot\frac{C}{4}$

$\textbf{To find:}$

$\text{The value of k}$

$\text{Consider,}$

$A+B+C=360^{\circ}$

$\dfrac{A}{4}+\dfrac{B}{4}+\dfrac{C}{4}=90^{\circ}$

$\dfrac{A}{4}+\dfrac{B}{4}=90^{\circ}-\dfrac{C}{4}$

$\implies\,tan(\dfrac{A}{4}+\dfrac{B}{4})=tan(90^{\circ}-\dfrac{C}{4})$

$\implies\,\dfrac{tan\frac{A}{4}+tan\frac{B}{4}}{1-tan\frac{A}{4}\,tan\frac{B}{4}}=cot\frac{C}{4}$

$\implies\,\dfrac{\frac{1}{cot\frac{A}{4}}+\frac{1}{cot\frac{B}{4}}}{1-\frac{1}{cot\frac{A}{4}}\,\frac{1}{cot\frac{B}{4}}}=cot\frac{C}{4}$

$\implies\,\dfrac{cot\frac{A}{4}+cot\frac{B}{4}}{cot\frac{A}{4}\,cot\frac{B}{4}-1}=cot\frac{C}{4}$

$\implies\,\cot\frac{A}{4}+\cot\frac{B}{4}+\cot\frac{C}{4}=\cot\frac{A}{4}\;\cot\frac{B}{4}\;\cot\frac{C}{4}$

$\textbf{But given,}$

$\cot\frac{A}{4}+\cot\frac{B}{4}+\cot\frac{C}{4}=k\,\cot\frac{A}{4}\;\cot\frac{B}{4}\;\cot\frac{C}{4}$

$\cot\frac{A}{4}\,\cot\frac{B}{4}\,\cot\frac{C}{4}=k\,\cot\frac{A}{4}\;\cot\frac{B}{4}\;\cot\frac{C}{4}$

$\implies\bf\,k=1$

$\therefore\textbf{The value of k is 1}$

Find more:

यदि A+B+C= 900 तो सिद्ध कीजिए: tanAtanB+tanBtanC+tanCtanA=1

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Answer 2

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