IF tanA - tanb = x and cot B - cotA = y, prove : cot (A-B) = x+y / xy

Asked by Sanjay | 13th May, 2015, 12:49: AM

Expert Answer:

$C o n s i d e r space t h e space g i v e n space e q u a t i o n s comma space tan A minus tan B equals x space a n d space c o t B minus c o t A equals y W e space n e e d space t o space p r o v e space t h a t space c o t open parentheses A minus B close parentheses equals fraction numerator x plus y over denominator x y end fraction N o w space c o n s i d e r space t h e space e q u a t i o n comma space tan A minus tan B equals x rightwards double arrow fraction numerator 1 over denominator c o t A end fraction minus fraction numerator 1 over denominator c o t B end fraction equals x space space space space space space space space space open square brackets sin c e space tan theta equals fraction numerator 1 over denominator c o t theta end fraction close square brackets rightwards double arrow fraction numerator c o t B minus c o t A over denominator c o t A c o t B end fraction equals x space space space space space space space rightwards double arrow fraction numerator y over denominator c o t A c o t B end fraction equals x space space space space space space space space space space space space space space space space space space open square brackets sin c e space c o t B minus c o t equals y close square brackets rightwards double arrow y over x equals c o t A c o t B rightwards double arrow c o t A c o t B equals y over x... left parenthesis 1 right parenthesis L. H. S equals c o t open parentheses A minus B close parentheses space space space space space space space space space space equals fraction numerator c o t A c o t B plus 1 over denominator c o t B minus c o t A end fraction space space space space space space space space space open square brackets sin c e space c o t open parentheses A minus B close parentheses space equals fraction numerator c o t A c o t B plus 1 over denominator c o t B minus c o t A end fraction space close square brackets space space space space space space space space space space equals fraction numerator y over x plus 1 over denominator y end fraction space space space space space space space space space space space space space space space space space space space open square brackets f r o m space g i v e n space a n d space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets space space space space space space space space space space equals fraction numerator fraction numerator x plus y over denominator x end fraction over denominator y end fraction space space space space space space space space space space space equals fraction numerator x plus y over denominator x y end fraction space space space space space space space space space space equals R. H. S. H e n c e space p r o v e d. space$

Answered by Vimala Ramamurthy | 13th May, 2015, 09:01: AM

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