A) \[4{{\sin }^{2}}\alpha -2{{x}^{2}}{{y}^{2}}\]
B) \[4{{\cos }^{2}}\alpha +2{{x}^{2}}{{y}^{2}}\]
C) \[4{{\sin }^{2}}\alpha \]
D) \[2{{\sin }^{2}}\alpha \]
Correct Answer: C
Solution :
\[{{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2}=\alpha \] \[cos({{\cos }^{-1}}x-{{\cos }^{-1}}\frac{y}{2})=\cos \alpha \] \[\Rightarrow x\times \frac{y}{2}+\sqrt{1-{{x}^{2}}}\sqrt{1-\frac{{{y}^{2}}}{4}}=\cos \alpha \] \[\Rightarrow {{\left( \cos \alpha -\frac{xy}{2} \right)}^{2}}=\left( 1-{{x}^{2}} \right)\left( 1-\frac{{{y}^{2}}}{4} \right)\] \[{{x}^{2}}+\frac{{{y}^{2}}}{4}-xy\cos \alpha =1-{{\cos }^{2}}\alpha ={{\sin }^{2}}\alpha \]
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