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In a triangle, tan(A)+tan(B)+tan(C) is equal to which of the following- a)1 b)0 c)infinite d)tanA+tanB+tanC

In a triangle, tan(A)+tan(B)+tan(C) is equal to which of the following-
 
a)1
b)0
c)infinite 
d)tanA+tanB+tanC

Grade:9

1 Answers

Riddhish Bhalodia
askIITians Faculty 434 Points
8 years ago
I am guessing the last option is actually tanAtanBtanC so I will prove that

as A,B,C are angles of a triangle
A+B+C = 180
thus
tanA = tan(180-B-C) = -tan(B+C) = -\frac{tanB+tanC}{1-tanBtanC}

now computing the sum of the three tangents
tanA +tanB+tanC = -\frac{tanB+tanC}{1-tanBtanC} + (tanB+tanC) = (tanB+tanC)(1 - \frac{1}{1-tanBtanC})

which results into
tanA +tanB+tanC = -tanBtanC\frac{tanB+tanC}{1-tanBtanC} = tanBtanCtanAHence Proved

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