0

Guest

Courses New

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

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

Grade:11

2 Answers

Arun
25750 Points
4 years ago
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
 
 

Think You Can Provide A Better Answer ?

Other Related Questions on Trigonometry

View all Questions »

ASK QUESTION

Get your questions answered by the expert for free

Answer and earn gift vouchers

Win Gift vouchers upto Rs 500/-

View courses by askIITians

Design classes One-on-One in your own way with Top IITians/Medical Professionals

Click Here Know More

Complete Self Study Package designed by Industry Leading Experts

Click Here Know More

Live 1-1 coding classes to unleash the Creator in your Child

Click Here Know More

a Complete All-in-One Study package Fully Loaded inside a Tablet!

Click Here Know More