In a triangle ABC,

(b^2 - c^2) / (b^2 + c^2) = sin(B - C) / sin(B + C), prove that it is either a right-angled or isoceles triangle.

Asked by kandappan | 13th Feb, 2020, 01:40: PM

Expert Answer:

It is given that
(b2 - c2)/(b2 + c2) = sin(B - C)/sin(B + C)
To prove that triangle ABC is either a right-angles or isosceles triangle.
fraction numerator b squared minus c squared over denominator b squared plus c squared end fraction equals fraction numerator sin left parenthesis B minus C right parenthesis over denominator sin left parenthesis B plus C right parenthesis end fraction
rightwards double arrow fraction numerator sin left parenthesis B plus C right parenthesis over denominator sin left parenthesis B C right parenthesis end fraction equals fraction numerator b squared plus c squared over denominator b squared minus c squared end fraction
rightwards double arrow fraction numerator sin left parenthesis B plus C right parenthesis plus sin left parenthesis B minus C right parenthesis over denominator sin left parenthesis B plus C right parenthesis minus sin left parenthesis B minus C right parenthesis end fraction equals fraction numerator b squared plus c squared plus b squared minus c squared over denominator b squared plus c squared minus b squared plus c squared end fraction space... space U sin g space c o m p o n e n d o space a n d space d i v i d e n d o
rightwards double arrow fraction numerator 2 sin B space cos C over denominator 2 cos B space sin C end fraction equals fraction numerator 2 b squared over denominator 2 c squared end fraction
N o w comma space fraction numerator a over denominator sin A end fraction equals fraction numerator b over denominator sin B end fraction equals fraction numerator c over denominator sin C end fraction equals 1 over k space... space S i n e space r u l e
rightwards double arrow fraction numerator b k space cos C over denominator c k space cos B end fraction equals fraction numerator 2 b squared over denominator 2 c squared end fraction
rightwards double arrow fraction numerator cos C over denominator cos B end fraction equals b over c
rightwards double arrow fraction numerator begin display style fraction numerator a squared plus b squared minus c squared over denominator 2 a b end fraction end style over denominator begin display style fraction numerator a squared plus c squared minus b squared over denominator 2 a c end fraction end style end fraction equals b over c
rightwards double arrow fraction numerator open parentheses a squared plus b squared minus c squared close parentheses c over denominator open parentheses a squared plus c squared minus b squared close parentheses b end fraction equals b over c
rightwards double arrow open parentheses a squared plus b squared minus c squared close parentheses c squared equals open parentheses a squared plus c squared minus b squared close parentheses b squared
rightwards double arrow a squared c squared plus b squared c squared minus c to the power of 4 equals a squared b squared plus b squared c squared minus b to the power of 4
rightwards double arrow a squared c squared minus a squared b squared equals c to the power of 4 minus b to the power of 4
rightwards double arrow a squared open parentheses c squared minus b squared close parentheses equals open parentheses c squared minus b squared close parentheses open parentheses c squared plus b squared close parentheses
rightwards double arrow open parentheses c squared minus b squared close parentheses open square brackets a squared minus open parentheses c squared plus b squared close parentheses close square brackets equals 0
rightwards double arrow c squared minus b squared equals 0 space space space o r space space a squared minus open parentheses c squared plus b squared close parentheses equals 0
rightwards double arrow b equals c space space space o r space space a squared equals c squared plus b squared
H e n c e comma space A B C thin space i s space e i t h e r space a space r i g h t minus a n g l e d space t r i a n g l e space o r space a n space i s o s c e l e s space t r i a n g l e.

Answered by Renu Varma | 13th Feb, 2020, 03:14: PM

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