Q. Prove the following identity, where the angles involved are acute angles for which the expressions are defined:-
 
cosA-sinA+1
------------ = cosecA + cotA, using the identity (cosec)square A=1+(cot)square A
cosA+sinA-1

Asked by dev680jai | 14th May, 2017, 10:54: AM

Expert Answer:

begin mathsize 16px style straight L. straight H. straight S. equals fraction numerator cosA minus sinA plus 1 over denominator cosA plus sinA minus 1 end fraction
equals fraction numerator cotA minus 1 plus cosecA over denominator cotA plus 1 minus cosecA end fraction space space space space space space space space space space left parenthesis Dividing space throughout space by space sinA right parenthesis
equals fraction numerator cotA plus cosecA minus 1 over denominator cotA minus cosecA plus 1 end fraction
equals fraction numerator left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosec squared straight A minus cot squared straight A right parenthesis over denominator cotA minus cosecA plus 1 end fraction space space space space space space space space left parenthesis cosec squared straight A equals 1 plus cot squared straight A right parenthesis
equals fraction numerator begin display style left parenthesis cotA plus cosecA right parenthesis minus left parenthesis cosecA minus cotA right parenthesis left parenthesis cosecA plus cotA right parenthesis end style over denominator begin display style cotA minus cosecA plus 1 end style end fraction
equals fraction numerator begin display style left parenthesis end style cos begin display style ecA end style begin display style plus end style begin display style cotA end style begin display style right parenthesis end style begin display style left square bracket end style begin display style 1 end style begin display style minus end style begin display style cosecA end style begin display style plus end style begin display style cotA end style begin display style right square bracket end style over denominator left parenthesis 1 minus cosecA plus cotA right parenthesis end fraction
equals cosecA plus cotA
equals straight R. straight H. straight S. end style

Answered by Rashmi Khot | 15th May, 2017, 08:49: AM