Question

Tan 25 tan 31 + tan 31 tan 34 + tan 34 + yan 25

Answer 1

$\textbf{To find:}$

$\text{The value of tan25^{\circ}\,tan31^{\circ}+tan31^{\circ}\,tan34^{\circ}+tan34^{\circ}\,tan25^{\circ} }$

$\textbf{Solution:}$

$\text{We know that,}$

$tan90^{\circ}=\infty$

$tan(25^{\circ}+65^{\circ})=\infty$

$\text{Using the following identity}$

$\boxed{\bf\,tan(A+B)=\frac{tanA+tanB}{1-tanA\,tanB}}$

$\dfrac{tan25^{\circ}+tan65^{\circ}}{1-tan25^{\circ}\,tan65^{\circ}}=\infty$

$\implies\,1-tan25^{\circ}\,tan65^{\circ}=0$

$\implies\,1-tan25^{\circ}\,tan(31^{\circ}+34^{\circ})=0$

$\implies\,1-tan25^{\circ}\,(\dfrac{tan31^{\circ}+tan34^{\circ}}{1-tan31^{\circ}\,tan34^{\circ}})=0$

$\implies\,1-tan31^{\circ}\,tan34^{\circ}-tan25^{\circ}\,(tan31^{\circ}+tan34^{\circ})=0$

$\implies\,1-tan31^{\circ}\,tan34^{\circ}-tan25^{\circ}\,tan31^{\circ}-tan25^{\circ}\,tan34^{\circ})=0$

$\text{Rearranging terms, we get}$

$\therefore\bf\,tan25^{\circ}\,tan31^{\circ}+tan31^{\circ}\,tan34^{\circ}+tan34^{\circ}\,tan25^{\circ}=1$

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यदि A+B+C= 90 तो सिद्ध कीजिए: tanAtanB+tanBtanC+tanCtanA=1

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