How Do You Rationalize The Numerator For √cosxsinx - Vedantu

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AnswerQuestion Answers for Class 12Class 12 BiologyClass 12 ChemistryClass 12 EnglishClass 12 MathsClass 12 PhysicsClass 12 Social ScienceClass 12 Business StudiesClass 12 EconomicsQuestion Answers for Class 11Class 11 EconomicsClass 11 Computer ScienceClass 11 BiologyClass 11 ChemistryClass 11 EnglishClass 11 MathsClass 11 PhysicsClass 11 Social ScienceClass 11 AccountancyClass 11 Business StudiesQuestion Answers for Class 10Class 10 ScienceClass 10 EnglishClass 10 MathsClass 10 Social ScienceClass 10 General KnowledgeQuestion Answers for Class 9Class 9 General KnowledgeClass 9 ScienceClass 9 EnglishClass 9 MathsClass 9 Social ScienceQuestion Answers for Class 8Class 8 ScienceClass 8 EnglishClass 8 MathsClass 8 Social ScienceQuestion Answers for Class 7Class 7 ScienceClass 7 EnglishClass 7 MathsClass 7 Social ScienceQuestion Answers for Class 6Class 6 ScienceClass 6 EnglishClass 6 MathsClass 6 Social ScienceQuestion Answers for Class 5Class 5 ScienceClass 5 EnglishClass 5 MathsClass 5 Social ScienceQuestion Answers for Class 4Class 4 ScienceClass 4 EnglishClass 4 MathsHow do you rationalize the numerator for \[\sqrt{\dfrac{\cos x}{\sin x}}\]?AnswerVerified554.1k+ viewsHint: In elementary algebra, root rationalisation is a process by which radicals in the denominator or numerator of an algebraic fraction are eliminated. To rationalize a term we need to multiply it with a term such that the square root, cube root etc is eliminated. We multiply with the same term in denominator and numerator to maintain the original value of the fraction which is to be rationalized.Complete step by step answer:Here we have the fraction \[\sqrt{\dfrac{\cos x}{\sin x}}\] where we need to rationalize the numerator.In fraction we have two term numerator and denominator. The term above the line is called numerator and the term below the line is called denominator. For example in the fraction \[\dfrac{2}{3}\], the above term that is 2 is numerator and the below term, 3, is the denominator.Thus we can clearly see in the term \[\sqrt{\dfrac{\cos x}{\sin x}}\] the numerator is \[\sqrt{\cos x}\] and denominator is \[\sqrt{\sin x}\]Now to rationalize the numerator we need to multiply it with a term such that the square root is removed. As we know multiplying two same square root terms result in same term , we will multiply both denominator and numerator by \[\sqrt{\cos x}\]\[\Rightarrow \sqrt{\dfrac{\cos x}{\sin x}}\left( \dfrac{\sqrt{\cos x}}{\sqrt{\cos x}} \right)\](We know that \[\dfrac{\sqrt{\cos x}}{\sqrt{\cos x}}=1\] and multiplying anything with 1 will not alter its value)\[\Rightarrow \sqrt{\dfrac{\cos x(\cos x)}{\sin x(\cos x)}}\]\[\Rightarrow \sqrt{\dfrac{{{(\cos x)}^{2}}}{\sin x(\cos x)}}\]\[\Rightarrow \dfrac{\sqrt{{{(\cos x)}^{2}}}}{\sqrt{\sin x(\cos x)}}\] , (by using the property \[\Rightarrow \sqrt{\dfrac{a}{b}}\Leftrightarrow \dfrac{\sqrt{a}}{\sqrt{b}}\] )\[\Rightarrow \dfrac{(\cos x)}{\sqrt{\sin x\cos x}}\] , (by using the property \[\Rightarrow \sqrt{{{a}^{2}}}=a\] )Hence the rationalized term required is \[\dfrac{(\cos x)}{\sqrt{\sin x\cos x}}\] , (only numerator rationalized).Note: The rationalization of a fraction or a term is done to simplify it and to find out the actual value of the term in a more simple form. Generally the rationalization is done for the denominator, unlike in the above question, to make the denominator the same/rational in case there are multiple terms in an expression.Recently Updated PagesMaster Class 10 Computer Science: Engaging Questions & Answers for SuccessMaster Class 10 General Knowledge: Engaging Questions & Answers for SuccessMaster Class 10 English: Engaging Questions & Answers for SuccessMaster Class 10 Social Science: Engaging Questions & Answers for SuccessMaster Class 10 Maths: Engaging Questions & Answers for SuccessMaster Class 10 Science: Engaging Questions & Answers for SuccessMaster Class 10 Computer Science: Engaging Questions & Answers for SuccessMaster Class 10 General Knowledge: Engaging Questions & Answers for SuccessMaster Class 10 English: Engaging Questions & Answers for SuccessMaster Class 10 Social Science: Engaging Questions & Answers for SuccessMaster Class 10 Maths: Engaging Questions & Answers for SuccessMaster Class 10 Science: Engaging Questions & Answers for Success

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