In a triangle ABC,prove that tan B-C/2=b-c/b+c  .cot A/2.

Asked by Ravikant | 26th Nov, 2015, 02:33: PM

Expert Answer:

begin mathsize 14px style T o space p r o v e colon space tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses equals open parentheses fraction numerator b minus c over denominator b plus c end fraction close parentheses c o t A over 2 P r o o f colon L e t 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 k T h e n comma a equals k sin A comma b equals k sin B comma c equals k sin C R. H. S. equals open parentheses fraction numerator b minus c over denominator b plus c end fraction close parentheses c o t A over 2 equals open parentheses fraction numerator k sin B minus k sin C over denominator k sin B plus k sin C end fraction close parentheses c o t A over 2 equals open parentheses fraction numerator sin B minus sin C over denominator sin B plus sin C end fraction close parentheses c o t A over 2 equals open square brackets fraction numerator 2 sin open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses cos open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses over denominator 2 sin open parentheses begin display style fraction numerator B plus C over denominator 2 end fraction end style close parentheses cos open parentheses begin display style fraction numerator B minus C over denominator 2 end fraction end style close parentheses end fraction close square brackets c o t A over 2 equals tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses c o t open parentheses fraction numerator B plus C over denominator 2 end fraction close parentheses c o t A over 2 equals tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses c o t open parentheses straight pi over 2 minus A over 2 close parentheses c o t A over 2 equals tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses tan A over 2 c o t A over 2 equals tan open parentheses fraction numerator B minus C over denominator 2 end fraction close parentheses equals L. H. S. end style

Answered by Rashmi Khot | 27th Nov, 2015, 08:42: AM