Question

$If space straight x equals cost left parenthesis 3 minus 2 cos squared straight t right parenthesis space and space straight y equals sint left parenthesis 3 minus 2 sin squared straight t right parenthesis comma space find space the space value space of space dy over dx at space straight t equals straight pi over 4.$

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Solution

straight x equals cost space left parenthesis 3 minus 2 cos squared straight t right parenthesis and straight y equals sint left parenthesis 3 minus sin squared straight t right parenthesis We space need space to space find space dy over dx colon dy over dx equals fraction numerator begin display style dy over dx end style over denominator begin display style dx over dt end style end fraction Let space us space find space dx over dt colon straight x equals cost left parenthesis 3 minus 2 cos squared straight t right parenthesis dx over dt equals cost left parenthesis 4 cost space sint right parenthesis plus left parenthesis 3 minus 2 cos squared straight t right parenthesis left parenthesis negative sint right parenthesis rightwards double arrow dx over dt equals negative 3 sint plus 4 cos squared tsint plus 2 cos squared tsint Let space us space find space dy over dt colon straight y equals sint left parenthesis 3 minus 2 sin squared straight t right parenthesis dy over dt equals sint left parenthesis negative 4 sint space cost right parenthesis plus left parenthesis 3 minus 2 sin squared straight t right parenthesis space left parenthesis cost right parenthesis rightwards double arrow dy over dt equals 3 cost minus 4 sin squared tcost minus 2 sin squared tcost Thus comma space dy over dx equals fraction numerator 3 cost minus 4 sin squared tcost minus 2 sin squared tcost over denominator negative 3 sint plus 4 cos squared tsint plus 2 cos squared tsint end fraction rightwards double arrow space dy over dx equals fraction numerator 3 cost minus 6 sin squared tcost over denominator negative 3 sint plus 6 cos squared tsint end fraction rightwards double arrow dy over dx equals fraction numerator 3 cost left parenthesis 1 minus 2 sin squared straight t right parenthesis over denominator negative 3 sint left parenthesis 1 minus 2 cos squared straight t right parenthesis end fraction rightwards double arrow dy over dx equals fraction numerator 3 cost left parenthesis 1 minus 2 sin squared straight t right parenthesis over denominator 3 sint left parenthesis 2 cos squared straight t minus 1 right parenthesis end fraction rightwards double arrow dy over dx equals fraction numerator cost space over denominator sint end fraction open square brackets because 2 cos squared straight t minus 1 equals 1 minus 2 sin squared straight t close square brackets rightwards double arrow dy over dx equals cot space straight t rightwards double arrow open parentheses dy over dx close parentheses subscript straight t equals straight pi over 4 end subscript equals cot straight pi over 4 equals 1

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