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Grade: 11

                        

If A+B+C=pi Show that tanA=tanB+tanC If (cot(alpha-beta)/ cot alpha ) +(cos^2gama/cos^2 alpha)=1

6 months ago

Answers : (2)

Arun
24742 Points
							
3 tanФ = Cot Ф = 1/tanФ
3 tan²Ф = 1
TanФ = + 1/√3 
Ф = n π + π/6
Ф = π/6 or π+π/6 = 7π/6 or π-π/6 = 5π/6 or 2π-π/6 = 11π/6
 
There are 4 solutions for 0°
===============
 
A + B + C = π/2
  => Tan(A+B) = Tan(π/2-C) = Cot C = 1/TanC
  => Tan (B+C) = 1/TanA
  => Tan(C+A) = 1/TanB
 
Tan (A+B) = [TanA + TanB]/[1 - TanA TanB]
Rearrange:  TanA TanB = 1 - (TanA+TanB)/Tan(A+B)
                                      = 1 - (TanA+TAnB) ×TanC     ---(1)
 
Similarly:   Tan B TanC = 1 - (TanB + TanC) × TanA  ---(2)
                  Tan C TanA = 1 - (TanC + TanA) ×TanB  --- (3)
 
Add (1) , (2) & (3):
LHS = TanA TanB + TanB TanC + TanC Tan A
        = 3 - 2 (TanA TanB + TanB TanC + Tanc TanA)
        = 3 -  2 LHS
 
So  TanA TanB + TanB TanC + Tan C TanA = 1
 
 
6 months ago
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