A) \[{{X}^{3}}+{{Y}^{3}}=1\]
B) \[{{X}^{2}}-{{Y}^{2}}=1\]
C) \[{{X}^{2}}+{{Y}^{2}}=1\]
D) \[{{X}^{4}}+{{Y}^{4}}=1\]
Correct Answer: C
Solution :
\[X{{\sin }^{3}}\theta +Y{{\cos }^{3}}\theta =\sin \theta \cos \theta \] ?..(i) \[X\sin \theta =Y\cos \theta \] ?..(ii) Using (ii) in (i), we get \[\Rightarrow \]\[Y\cos \theta {{\sin }^{2}}\theta +Y{{\cos }^{3}}\theta =\sin \theta \cos \theta \] \[\Rightarrow \] \[Y{{\sin }^{2}}\theta +Y{{\cos }^{2}}\theta =\sin \theta \,\,\Rightarrow \,\,Y=\sin \theta \] \[\therefore \] \[X\sin \theta =\sin \theta \times \cos \theta \Rightarrow X=\cos \theta \] \[\therefore \] \[{{X}^{2}}+{{Y}^{2}}=1\]
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