If a s e c theta space plus space b tan theta space plus space c space equals space 0 and p s e c theta space plus space q tan theta space plus space r space equals space 0, prove that left parenthesis b r minus q c right parenthesis squared space minus space left parenthesis p c minus a r right parenthesis squared space equals space left parenthesis a q minus b p right parenthesis squared

Asked by Jerlin George | 2nd Oct, 2015, 10:57: AM

Expert Answer:

C o n s i d e r space t h e space s y s t e m space o f space e q u a t i o n s a s e c theta plus b tan theta plus c equals 0... left parenthesis 1 right parenthesis p s e c theta plus q tan theta plus r equals 0... left parenthesis 2 right parenthesis  M u l t i p l y i n g space t h e space e q u a t i o n space left parenthesis 1 right parenthesis space b y space p space a n d space e q u a t i o n space left parenthesis 2 right parenthesis space b y space a comma space w e space h a v e comma a p s e c theta plus b p t a n theta equals negative p c... left parenthesis 3 right parenthesis a p s e c theta plus a q t a n theta equals negative a r... left parenthesis 4 right parenthesis S u b t r a c t i n g space e q u a t i o n space left parenthesis 4 right parenthesis space f r o m space left parenthesis 3 right parenthesis comma space w e space h a v e comma open parentheses b p minus a q close parentheses tan theta equals negative c p plus a r rightwards double arrow open parentheses b p minus a q close parentheses t a n theta equals a r minus c p rightwards double arrow t a n theta equals fraction numerator a r minus c p over denominator open parentheses b p minus a q close parentheses end fraction... left parenthesis 5 right parenthesis  M u l t i p l y i n g space t h e space e q u a t i o n space left parenthesis 1 right parenthesis space b y space q space a n d space e q u a t i o n space left parenthesis 2 right parenthesis space b y space b comma space w e space h a v e comma a q s e c theta plus b q t a n theta equals negative c q... left parenthesis 6 right parenthesis b p s e c theta plus b q t a n theta equals negative b r... left parenthesis 7 right parenthesis S u b t r a c t i n g space e q u a t i o n space left parenthesis 7 right parenthesis space f r o m space left parenthesis 6 right parenthesis comma space w e space h a v e comma open parentheses a q minus b p close parentheses s e c theta equals b r minus c q rightwards double arrow open parentheses a q minus b p close parentheses s e c theta equals b r minus c q rightwards double arrow s e c theta equals fraction numerator b r minus c q over denominator open parentheses a q minus b p close parentheses end fraction... left parenthesis 8 right parenthesis  W e space k n o w space t h a t comma space s e c squared theta minus tan squared theta equals 1 T h u s comma space s u b t i t u t i n g space t h e space v a l u e s space o f space s e c theta space a n d space tan theta space f r o m space e q u a t i o n s space left parenthesis 8 right parenthesis space a n d space left parenthesis 5 right parenthesis space i n space t h e space a b o v e space i d e n t i t y comma space w e space h a v e comma rightwards double arrow open square brackets fraction numerator b r minus c q over denominator open parentheses a q minus b p close parentheses end fraction close square brackets squared minus open square brackets fraction numerator a r minus c p over denominator open parentheses b p minus a q close parentheses end fraction close square brackets squared equals 1 rightwards double arrow open square brackets fraction numerator b r minus c q over denominator open parentheses a q minus b p close parentheses end fraction close square brackets squared minus open square brackets negative fraction numerator a r minus c p over denominator open parentheses a q minus b p close parentheses end fraction close square brackets squared equals 1 rightwards double arrow open square brackets b r minus c q close square brackets squared minus open square brackets c p minus a r close square brackets squared equals open parentheses a q minus b p close parentheses squared

Answered by Vimala Ramamurthy | 2nd Oct, 2015, 09:13: PM

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