Question : If $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$ and $x\sin\theta-y\cos\theta=0$, then the value of $\left ( x^{2}+y^{2} \right )$ equals:
Option 1: $1$
Option 2: $\frac{1}{2}$
Option 3: $\frac{3}2$
Option 4: $2$
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Correct Answer: $1$
Solution :
Given: $x\sin\theta=y\cos\theta$
And, $x\sin^{3}\theta +y\cos^{3}\theta=\sin\theta\cos\theta$
⇒ $x\sin\theta(\sin^{2}\theta) +y\cos\theta(\cos^{2}\theta)=\sin\theta\cos\theta$
Since, $x\sin\theta=y\cos\theta$ and $\sin^{2}\theta+\cos^{2}\theta=1$,
So, $x\sin\theta=y\cos\theta=\sin\theta\cos\theta$
⇒ $x=\cos\theta$ and $y=\sin\theta$
So, $x^{2}+y^{2}=\sin^{2}\theta+\cos^{2}\theta=1$
Hence, the correct answer is 1.
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