જો A + B + C = {180^o}, તો cot frac{A}{2} + cot frac{B}{2} + cot frac{C}{2} = . . .

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3.Trigonometrical Ratios, Functions and Identities

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જો $A + B + C = {180^o},$ તો $\cot \frac{A}{2} + \cot \frac{B}{2} + \cot \frac{C}{2} = . . .$

$2\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

$4\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

$\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

$8\,\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}$

Solution

$\therefore \,\frac{A}{2} + \frac{B}{2} = {90^o} – \frac{C}{2}$

$\therefore \cot \left( {\frac{A}{2} + \frac{B}{2}} \right) = \cot \left( {{{90}^o} – \frac{C}{2}} \right)$

or $\frac{{\cot \frac{A}{2}\,.\,\cot \frac{B}{2}\, – 1}}{{\cot \frac{B}{2}\, + \,\cot \frac{A}{2}}} = \tan \frac{C}{2} = \frac{1}{{\cot \frac{C}{2}}}$

or $\left( {\cot \frac{A}{2}\cot \frac{B}{2} – 1} \right)\cot \frac{C}{2} = \cot \frac{B}{2} + \cot \frac{A}{2}$

$\cot \frac{A}{2}\,.\,\cot \frac{B}{2}\,.\,\cot \frac{C}{2} = \cot \frac{C}{2} + \cot \frac{B}{2}$ $ + \cot \frac{A}{2}.$

Standard 11

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(IIT-1986)

જો $a{\sin ^2}x + b{\cos ^2}x = c,\,\,$$b\,{\sin ^2}y + a\,{\cos ^2}y = d$ અને $a\,\tan x = b\,\tan y,$ તો $\frac{{{a^2}}}{{{b^2}}} = . . ..$