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in a traingle ABC find the value of ∑tanb/2 . tanc/2

in a traingle ABC find the value of ∑tanb/2 . tanc/2 

Grade:12th pass

1 Answers

BALAJI ANDALAMALA
askIITians Faculty 78 Points
5 years ago

In a triangle ABC , A+B+C=180
\frac{A+B+C}{2}=90
Take Tan on both sides ,we get
tan(\frac{A}{2}+\frac{B}{2}) = tan(90-\frac{C}{2}) = cot\frac{C}{2}
Apply the formula tan(x+y) = (tanx+tany)/(1-tanxtany) ,we get
\Rightarrow \frac{tan\frac{A}{2}+tan\frac{B}{2}}{1-tan\frac{A}{2}tan\frac{B}{2}} =cot\frac{C}{2}
\Rightarrow tan\frac{A}{2}tan\frac{C}{2}+tan\frac{B}{2}tan\frac{C}{2}={1-tan\frac{A}{2}tan\frac{B}{2}}
\therefore tan\frac{A}{2}tan\frac{C}{2}+tan\frac{B}{2}tan\frac{C}{2} +tan\frac{A}{2}tan\frac{B}{2} =1

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