I F space tan to the power of minus 1 end exponent open parentheses fraction numerator x squared space minus space y squared over denominator x to the power of 2 space end exponent plus space y squared end fraction close parentheses space equals space a comma space p r o v e space t h a t fraction numerator space d y over denominator d x end fraction equals fraction numerator space x open parentheses 1 space minus space tan a close parentheses over denominator y open parentheses 1 space plus space tan a close parentheses end fraction

Asked by inderjeet.saluja | 23rd Aug, 2014, 10:51: PM

Expert Answer:

G i v e n tan to the power of minus 1 end exponent open parentheses fraction numerator x squared minus y squared over denominator x squared plus y squared end fraction close parentheses equals a rightwards double arrow fraction numerator x squared minus y squared over denominator x squared plus y squared end fraction equals tan space a  A p p l y i n g space c o m p o n e n d o space a n d space d i v i d e n d o comma space w e space g e t fraction numerator open parentheses x squared minus y squared close parentheses minus open parentheses x squared plus y squared close parentheses over denominator open parentheses x squared minus y squared close parentheses plus open parentheses x squared plus y squared close parentheses end fraction equals fraction numerator tan space a minus 1 over denominator tan space a space plus 1 end fraction  fraction numerator minus 2 y squared over denominator 2 x squared end fraction equals fraction numerator tan space a minus 1 over denominator tan space a space plus 1 end fraction  y squared over x squared equals fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction space space space space space minus minus minus minus minus minus 1  y equals x squared over y open parentheses fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction close parentheses  fraction numerator d y over denominator d x end fraction equals open parentheses fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction close parentheses open square brackets fraction numerator 2 x y minus x squared begin display style fraction numerator d y over denominator d x end fraction end style over denominator y squared end fraction close square brackets  f r o m space 1 space w e space h a v e space 1 over x squared equals fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction cross times 1 over y squared  fraction numerator d y over denominator d x end fraction equals 1 over x squared open square brackets 2 x y minus x squared fraction numerator d y over denominator d x end fraction close square brackets  2 fraction numerator d y over denominator d x end fraction equals fraction numerator 2 y over denominator x end fraction  fraction numerator d y over denominator d x end fraction equals y over x  f r o m space 1 space w e space h a v e comma space y over x equals x over y open parentheses fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction close parentheses  therefore fraction numerator d y over denominator d x end fraction equals y over x equals x over y open parentheses fraction numerator 1 minus tan space a over denominator 1 plus tan space a end fraction close parentheses  H e n c e space P r o v e d

Answered by Dharma Teja | 25th Aug, 2014, 10:26: AM

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