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Trigonometry> In a triangle, tan(A)+tan(B)+tan(C) is eq...

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

Prithvish Shaunkani , 9 Years ago

Grade 9

1 Answers

Riddhish Bhalodia

Last Activity: 9 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
$tanA +tanB+tanC = -tanBtanC\frac{tanB+tanC}{1-tanBtanC} = tanBtanCtanA$ Hence Proved

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