if nsinA=mcosA, PROVE THAT:
(msinA-ncosA/msinA+ncosA) (BOTH R ONE TERM, THEN NEW TERM IS)  + (msinA+ncosA/msinA+ncosA)=2(m to the power 4+ nto the power 4/m to the power4-n to the power 4)

Asked by smruti2002ranjan | 21st Sep, 2016, 11:56: PM

Expert Answer:

begin mathsize 16px style straight n space sin space straight A space equals space straight m space cos space straight A
rightwards double arrow fraction numerator sin space straight A over denominator cos space straight A end fraction equals straight m over straight n
rightwards double arrow tan space straight A space equals space straight m over straight n space space space space space space space.... left parenthesis 1 right parenthesis
Now comma
fraction numerator straight m space sin space straight A space minus space straight n space cos space straight A over denominator straight m space sin space straight A space plus space straight n space cos space straight A end fraction space plus space fraction numerator straight m space sin space straight A space plus space straight n space cos space straight A over denominator straight m space sin space straight A space minus space straight n space cos space straight A end fraction
equals space fraction numerator straight m space tan space straight A space minus space straight n over denominator straight m space tan space straight A space plus space straight n end fraction space plus space fraction numerator straight m space tan space straight A space plus space straight n over denominator straight m space tan space straight A space minus space straight n end fraction
equals fraction numerator straight m space cross times space begin display style straight m over straight n end style space minus space straight n over denominator straight m space cross times space straight m over straight n space plus space straight n end fraction space plus space fraction numerator straight m space cross times space straight m over straight n space plus space straight n over denominator straight m space cross times space straight m over straight n space minus space straight n end fraction
equals fraction numerator straight m squared space minus space straight n squared over denominator straight m squared space plus space straight n squared end fraction space plus space fraction numerator straight m squared space plus space straight n squared over denominator straight m squared space minus space straight n squared end fraction
equals fraction numerator open parentheses straight m squared space minus space straight n squared close parentheses squared space plus space open parentheses straight m squared space plus space straight n squared close parentheses squared over denominator straight m to the power of 4 space minus space straight n to the power of 4 end fraction
equals fraction numerator straight m to the power of 4 space minus space 2 straight m squared straight n squared space plus space straight n to the power of 4 space plus space straight m to the power of 4 space plus space 2 straight m squared straight n squared space plus space straight n to the power of 4 over denominator straight m to the power of 4 space minus space straight n to the power of 4 end fraction
equals fraction numerator straight m to the power of 4 space plus space straight n to the power of 4 space plus space straight m to the power of 4 space plus space straight n to the power of 4 over denominator straight m to the power of 4 space minus space straight n to the power of 4 end fraction
equals fraction numerator 2 space open parentheses straight m to the power of 4 space plus space straight n to the power of 4 space close parentheses over denominator straight m to the power of 4 space minus space straight n to the power of 4 end fraction
Hence comma space proved end style

Answered by Rashmi Khot | 22nd Sep, 2016, 10:46: AM