A) \[1\]
B) \[x/y\]
C) \[\frac{{{x}^{n-1}}}{{{y}^{n-1}}}\]
D) None of these
Correct Answer: C
Solution :
Put \[{{x}^{n}}=\cos \alpha ,\,\,{{y}^{n}}=\cos \beta \] \[\Rightarrow \,\,a=\frac{\sin \alpha +\sin \beta }{\cos \alpha -\cos \beta }=\frac{2\sin \left( \frac{\alpha +\beta }{2} \right)\cos \left( \frac{\alpha -\beta }{2} \right)}{-2\sin \left( \frac{\alpha +\beta }{2} \right)\sin \left( \frac{\alpha -\beta }{2} \right)}\]\[=-\cot \,\left( \frac{\alpha -\beta }{2} \right)\] \[\Rightarrow \,2{{\cot }^{-1}}(-a)=\alpha -\beta \] \[\Rightarrow \,\,{{\cos }^{-1}}({{x}^{n}})-{{\cos }^{-1}}({{y}^{n}})=2{{\cot }^{-1}}(-a)\] \[\Rightarrow \,\frac{{{y}^{n-1}}}{\sqrt{1-{{y}^{2n}}}}\frac{dy}{dx}=\frac{{{x}^{n-1}}}{\sqrt{1-{{x}^{2n}}}}\Rightarrow \sqrt{\frac{1-{{x}^{2n}}}{1-{{y}^{2n}}}\frac{dy}{dx}}=\frac{{{x}^{n-1}}}{{{y}^{n-1}}}\]
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