Please solve this question as fast as possible.I f space alpha space a n d space beta space a r e space t h e space r o o t s space o f space t h e space e q u a t i o n space a space tan ϑ space plus space b space s e c theta space equals c space a n d space tan left parenthesis alpha plus beta right parenthesis space equals space fraction numerator tan alpha space plus space tan beta over denominator 1 minus tan alpha tan beta end fraction comma space t h e n space p r o v e space t h a t space t a n left parenthesis alpha plus beta right parenthesis space equals space fraction numerator 2 a c over denominator a squared minus c squared end fraction

Asked by vatsalchoudhary41 | 24th Sep, 2015, 06:00: 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 comma a tan theta plus b s e c theta equals c rightwards double arrow a t a n theta minus c equals negative b s e c theta S q u a r i n g space b o t h space t h e space s i d e s comma space w e space h a v e comma rightwards double arrow open parentheses a tan theta minus c close parentheses squared equals open parentheses negative b s e c theta close parentheses squared rightwards double arrow a squared tan squared theta plus c squared minus 2 a c tan theta equals b squared s e c squared theta rightwards double arrow a squared t a n squared theta plus c squared minus 2 a c t a n theta equals b squared open parentheses 1 plus tan squared theta close parentheses rightwards double arrow a squared t a n squared theta plus c squared minus 2 a c t a n theta minus b squared minus b squared t a n squared theta equals 0 rightwards double arrow open parentheses a squared minus b squared close parentheses t a n squared theta minus 2 a c t a n theta plus c squared minus b squared equals 0 S i n c e space tan alpha space a n d space tan beta space a r e space t h e space r o o t s space o f space t h e space a b o v e space e q u a t i o n comma space t h e space s u m space o f space t h e space r o o t s comma space tan alpha plus tan beta equals fraction numerator negative open parentheses negative 2 a c close parentheses over denominator a squared minus b squared end fraction equals fraction numerator 2 a c over denominator a squared minus b squared end fraction... left parenthesis 1 right parenthesis a n d space p r o d u c t space o f space t h e space r o o t s comma space tan alpha cross times tan beta equals fraction numerator c squared minus b squared over denominator a squared minus b squared end fraction... left parenthesis 2 right parenthesis N o w space c o n s i d e r comma space tan open parentheses alpha plus beta close parentheses equals fraction numerator space t a n alpha plus t a n beta over denominator 1 minus space t a n alpha cross times t a n beta end fraction... left parenthesis 3 right parenthesis S u b s t i t u t i n g space t h e space v a l u e s space t a n alpha plus t a n beta space a n d space t a n alpha cross times t a n beta space f r o m space e q u a t i o n s space left parenthesis 1 right parenthesis space a n d space left parenthesis 2 right parenthesis comma space i n space e q u a t i o n space left parenthesis 3 right parenthesis comma space w e space h a v e comma space t a n open parentheses alpha plus beta close parentheses equals fraction numerator fraction numerator 2 a c over denominator a squared minus b squared end fraction over denominator 1 minus fraction numerator c squared minus b squared over denominator a squared minus b squared end fraction end fraction rightwards double arrow t a n open parentheses alpha plus beta close parentheses equals fraction numerator fraction numerator 2 a c over denominator a squared minus b squared end fraction over denominator fraction numerator a squared minus b squared minus c squared plus b squared over denominator a squared minus b squared end fraction end fraction rightwards double arrow t a n open parentheses alpha plus beta close parentheses equals fraction numerator 2 a c over denominator a squared minus c squared end fraction H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 24th Sep, 2015, 09:19: AM