Solve:

1)      (a2- b2 )Sin theta  + 2ab cos theta = a2 + b then prove that tan theta = a2 – b2 / 2ab 

Asked by tanishqa.rathod | 13th Aug, 2015, 09:03: PM

Expert Answer:

G i v e n space e q u a t i o n space i s space open parentheses a squared minus b squared close parentheses sin theta plus 2 a b cos theta equals a squared plus b squared W e space k n o w space t h a t space sin theta equals fraction numerator 2 tan begin display style theta over 2 end style over denominator 1 plus tan squared theta over 2 end fraction space a n d space cos theta equals fraction numerator 1 minus t a n squared theta over 2 over denominator 1 plus t a n squared theta over 2 end fraction T h u s comma space s u b s t i t u t i n g space t h e space a b o v e space v a l u e s space i n space t h e space g i v e n space e q u a t i o n comma space w e space h a v e comma open parentheses a squared minus b squared close parentheses fraction numerator 2 t a n begin display style theta over 2 end style over denominator 1 plus t a n squared theta over 2 end fraction plus 2 a b fraction numerator 1 minus t a n squared theta over 2 over denominator 1 plus t a n squared theta over 2 end fraction equals a squared plus b squared N o w space s u b s t i t u t i n g comma space t equals t a n theta over 2 comma space w e space h a v e comma open parentheses a squared minus b squared close parentheses fraction numerator 2 t over denominator 1 plus t squared end fraction plus 2 a b fraction numerator 1 minus t squared over denominator 1 plus t squared end fraction equals a squared plus b squared rightwards double arrow open parentheses a squared minus b squared close parentheses 2 t plus 2 a b open parentheses 1 minus t squared close parentheses equals open parentheses a squared plus b squared close parentheses open parentheses 1 plus t squared close parentheses S i m p l i f y i n g comma space w e space h a v e comma open parentheses a squared plus 2 a b plus b close parentheses t squared minus 2 open parentheses a squared minus b squared close parentheses t plus open parentheses a squared minus 2 a b plus b close parentheses equals 0 R e w r i t i n g space t h e space a b o v e space q u a d r a t i c comma space w e space h a v e comma open parentheses a plus b close parentheses squared t squared minus 2 open parentheses a squared minus b squared close parentheses t plus open parentheses a minus b close parentheses squared equals 0 rightwards double arrow open parentheses a plus b close parentheses squared t squared minus 2 open parentheses a plus b close parentheses open parentheses a minus b close parentheses t plus open parentheses a minus b close parentheses squared equals 0 rightwards double arrow open square brackets open parentheses a plus b close parentheses t minus open parentheses a minus b close parentheses close square brackets squared equals 0 space space space space space space space space open square brackets A squared minus 2 A B plus B squared equals open parentheses A minus B close parentheses squared close square brackets rightwards double arrow open square brackets open parentheses a plus b close parentheses t minus open parentheses a minus b close parentheses close square brackets equals 0 space rightwards double arrow open parentheses a plus b close parentheses t equals open parentheses a minus b close parentheses rightwards double arrow t equals fraction numerator a minus b over denominator a plus b end fraction space... left parenthesis 1 right parenthesis rightwards double arrow t a n theta over 2 equals space fraction numerator a minus b over denominator a plus b end fraction... left parenthesis 2 right parenthesis W e space k n o w space t h a t comma space fraction numerator 2 t over denominator 1 minus t squared end fraction equals tan theta comma space w h e r e comma space t equals space t a n theta over 2. N o w space c o n s i d e r space tan theta equals fraction numerator 2 t over denominator 1 minus t squared end fraction rightwards double arrow t a n theta equals fraction numerator 2 fraction numerator a minus b over denominator a plus b end fraction over denominator 1 minus open parentheses fraction numerator a minus b over denominator a plus b end fraction close parentheses squared end fraction space space space space space open square brackets f r o m space e q u a t i o n space left parenthesis 1 right parenthesis close square brackets rightwards double arrow t a n theta equals fraction numerator 2 open parentheses a plus b close parentheses open parentheses a minus b close parentheses over denominator open parentheses a plus b close parentheses squared minus open parentheses a minus b close parentheses squared end fraction rightwards double arrow t a n theta equals fraction numerator 2 open parentheses a squared minus b squared close parentheses over denominator 4 a b end fraction rightwards double arrow t a n theta equals fraction numerator open parentheses a squared minus b squared close parentheses over denominator 2 a b end fraction

Answered by Vimala Ramamurthy | 14th Aug, 2015, 10:19: AM