Question

in a triangle ABC,a/tanA+b/tanB+c/tanC=
with explanation​

Answer 1

Answer:

If A+B+C=npi (n being an integer). Show that tanA+tanB+tanC=tanA tanB tanC. Prove that a triangle ABC is equilateral if and only if tanA+tanB+tanC=3sqrt(3). ... In right angled triangle ABC, right angled at C, show that tanA+tanB=(c^2)/(ab).

Answer 2

Answer:

∆ABC is an equilateral triangle.

Given:

tanA/a= tanB/b= tanC/c----------------(1)  

Step-by-step explanation:

In ∆ABC ,

By sine rule  in ∆ABC , we have          

a/sinA=b/sinB=c/sinC= k(let).

∴Putting a=k sin A , b = k sin B and c=k sin C in eq. (1)

⇒tanA/(ksinA)= tanB/(ksinB)= tanC/(ksinC)

⇒ (sinA/cosA)/sinA=(sinB/cosB)/sinB = (sinC/cosC)/sinC

⇒ 1/cosA = 1/cosB =1/cosC.

⇒ secA= secB=secC

⇒ A = B = C = x(let)

But A+B+C=180° (Triangle sum property)

x+x+x=180°

3x=180° ⇒x= 180°/3=60°

∴ A = B = C = 60°

Thus , ∆ABC is an equilateral triangle.

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