Question

If sin y=x sin(a+y), then prove that $dy over dx equals fraction numerator sin squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction$

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Solution

because space space space sin space straight y equals straight x space sin left parenthesis straight a plus straight y right parenthesis therefore space space space space space space space space space space straight x equals fraction numerator sin space straight y over denominator sin left parenthesis straight a plus straight y right parenthesis end fraction Differentiating space both space sides space straight w. straight r. straight t. space straight y. space space space space space space space space space dx over dy equals fraction numerator sin left parenthesis straight a plus straight y right parenthesis begin display style straight d over dy end style left parenthesis sin space straight y right parenthesis minus sin space straight y begin display style straight d over dy end style left square bracket sin left parenthesis straight a plus straight y right parenthesis right square bracket over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction space space space space space space space space space space space space space space space space space equals fraction numerator sin left parenthesis straight a plus straight y right parenthesis cos space straight y minus sin space straight y. cos left parenthesis straight a plus straight y right parenthesis over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction equals fraction numerator sin left square bracket left parenthesis straight a plus straight y right parenthesis minus straight y right square bracket over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction equals fraction numerator sin space straight a over denominator sin squared left parenthesis straight a plus straight y right parenthesis end fraction Now space dy over dx equals fraction numerator 1 over denominator begin display style dx over dy end style end fraction equals fraction numerator sin squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction

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Continuity And Differentiability

If sin y=x sin(a+y), then prove that $dy over dx equals fraction numerator sin squared left parenthesis straight a plus straight y right parenthesis over denominator sin space straight a end fraction$

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