Question

$If space cos to the power of negative 1 end exponent straight x over straight a plus cos to the power of negative 1 end exponent straight y over straight b equals straight a comma space space prove space that space straight x squared over straight a squared minus fraction numerator 2 space straight x space straight y over denominator straight a space straight b end fraction cos space straight a plus straight y squared over straight b squared equals sini squared straight a.$

Solution

because space space cos to the power of negative 1 end exponent straight x over straight a plus cos to the power of negative 1 end exponent straight x over straight b equals straight a therefore space space space cos to the power of negative 1 end exponent open square brackets straight x over straight a. straight y over straight b minus square root of open parentheses 1 minus straight x squared over straight a squared close parentheses open parentheses 1 minus straight x squared over straight a squared close parentheses end root close square brackets equals straight a rightwards double arrow space space space space space fraction numerator straight x space straight y over denominator straight a space straight b end fraction minus square root of open parentheses 1 minus straight x squared over straight a squared close parentheses open parentheses 1 minus straight x squared over straight a squared close parentheses end root equals cos space straight a space space space rightwards double arrow space space fraction numerator straight x space straight y over denominator straight a space straight b end fraction minus cos space straight a equals square root of open parentheses 1 minus straight x squared over straight a squared close parentheses open parentheses 1 minus straight x squared over straight a squared close parentheses end root rightwards double arrow space space space space open parentheses fraction numerator straight x space straight y over denominator straight a space straight b end fraction minus cos space straight a close parentheses squared equals open parentheses 1 minus straight x squared over straight a squared close parentheses open parentheses 1 minus straight y squared over straight b squared close parentheses rightwards double arrow space space space space fraction numerator straight x squared space straight y squared over denominator straight a squared space straight b squared end fraction minus fraction numerator 2 space straight x space straight y space over denominator straight a space straight b end fraction cos space straight a plus cos squared straight a equals 1 minus straight x squared over straight a squared minus straight y squared over straight b squared plus fraction numerator straight x squared space straight y squared over denominator straight a squared space straight b squared end fraction rightwards double arrow space fraction numerator straight x squared space straight y squared over denominator straight a squared space straight b squared end fraction minus fraction numerator 2 space straight x space straight y over denominator straight a space straight b end fraction cos space straight a plus cos squared space straight a equals 1 minus straight x squared over straight a squared minus straight y squared over straight b squared plus fraction numerator bold x to the power of bold 2 bold space bold y to the power of bold 2 over denominator bold a to the power of bold 2 bold space bold b to the power of bold 2 end fraction rightwards double arrow space space straight x squared over straight a squared minus fraction numerator 2 xy over denominator ab end fraction cos space straight a plus straight y squared over straight b squared equals 1 minus cos squared space straight a rightwards double arrow space space space straight x squared over straight a squared minus fraction numerator 2 xy over denominator ab end fraction cos space straight a plus straight y squared over straight b squared equals sin squared space straight a

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