If tan(A-B)/tanA+sin^C/sin^A =1 show that tanAtanB = tan^2C

Asked by Sanjay | 13th May, 2015, 09:23: PM

Expert Answer:

Q u e s t i o n space i s space i n c o r r e c t. space P l e a s e space c h e c k space t h e space q u e s t i o n. Q u e s t i o n space s h o u l d space b e colon I f space fraction numerator tan open parentheses A minus B close parentheses over denominator tan A end fraction plus fraction numerator sin squared C over denominator sin squared A end fraction equals comma space s h o w space t h a t space tan A cross times tan B equals tan squared C 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 colon fraction numerator tan open parentheses A minus B close parentheses over denominator tan A end fraction plus fraction numerator sin squared C over denominator sin squared A end fraction equals 1 rightwards double arrow fraction numerator fraction numerator tan open parentheses A minus B close parentheses cross times sin squared A over denominator tan A end fraction over denominator sin squared A end fraction plus fraction numerator sin squared C over denominator sin squared A end fraction equals 1 rightwards double arrow fraction numerator fraction numerator tan open parentheses A minus B close parentheses cross times sin squared A over denominator tan A end fraction plus sin squared C over denominator sin squared A end fraction equals 1 rightwards double arrow fraction numerator tan open parentheses A minus B close parentheses cross times sin squared A over denominator tan A end fraction plus sin squared C equals sin squared A rightwards double arrow sin squared C equals sin squared A minus fraction numerator tan open parentheses A minus B close parentheses cross times sin squared A over denominator tan A end fraction rightwards double arrow sin squared C equals sin squared A minus tan open parentheses A minus B close parentheses cross times sin A cos A... left parenthesis 1 right parenthesis rightwards double arrow 1 minus cos squared C equals sin squared A minus tan open parentheses A minus B close parentheses cross times sin A cos A rightwards double arrow cos squared C equals 1 minus sin squared A plus tan open parentheses A minus B close parentheses cross times sin A cos A rightwards double arrow cos squared C equals cos squared A plus tan open parentheses A minus B close parentheses cross times sin A cos A... left parenthesis 2 right parenthesis D i v i d i n g space e q u a t i o n space left parenthesis 1 right parenthesis space b y space left parenthesis 2 right parenthesis comma space w e space h a v e comma fraction numerator sin squared C over denominator cos squared C end fraction equals fraction numerator sin squared A minus tan open parentheses A minus B close parentheses cross times sin A cos A over denominator cos squared A plus tan open parentheses A minus B close parentheses cross times sin A cos A end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A minus tan open parentheses A minus B close parentheses cross times sin A cos A over denominator cos squared A plus tan open parentheses A minus B close parentheses cross times sin A cos A end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A minus begin display style fraction numerator tan A minus tan B over denominator 1 plus tan A tan B end fraction end style cross times sin A cos A over denominator cos squared A plus fraction numerator tan A minus tan B over denominator 1 plus tan A tan B end fraction cross times sin A cos A end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A cross times open parentheses 1 plus tan A tan B close parentheses minus begin display style open parentheses tan A minus tan B close parentheses end style cross times sin A cos A over denominator cos squared A cross times open parentheses 1 plus tan A tan B close parentheses plus open parentheses tan A minus tan B close parentheses cross times sin A cos A end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A plus sin squared A tan A tan B minus sin squared A plus sin A cos A tan B over denominator cos squared A plus sin A cos A tan B plus sin squared A minus sin A cos A tan B end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A tan A tan B plus sin A cos A tan B over denominator cos squared A plus sin squared A end fraction rightwards double arrow tan squared C equals fraction numerator sin squared A tan A tan B plus cos squared A tan A tan B over denominator cos squared A plus sin squared A end fraction space space space space space space space open square brackets m u l t i p l y space a n d space d i v i d e d space b y space cos space A space t o space t h e space t e r m space sin A cos A tan B close square brackets rightwards double arrow tan squared C equals sin squared A tan A tan B plus cos squared A tan A tan B rightwards double arrow tan squared C equals tan A tan B open parentheses sin squared A plus cos squared A close parentheses rightwards double arrow tan squared C equals tan A tan B H e n c e space p r o v e d.

Answered by Vimala Ramamurthy | 14th May, 2015, 10:26: AM