Question : If $x \sin 45° = y\operatorname{cosec} 30°$, then $\frac{x^4}{y^4}$ is equal to:
Option 1: $4^3$
Option 2: $6^3$
Option 3: $2^3$
Option 4: $8^3$
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Correct Answer: $4^3$
Solution :
Given: If $x \sin \ 45° = y \operatorname{cosec} 30°$.
$⇒\frac{x}{y} = \frac{\operatorname{cosec}30°}{\sin 45°}$
Substitute the values of the given trigonometric ratios, we get,
$⇒\frac{x}{y} = \frac{2}{\frac{1}{\sqrt{2}}}$
$⇒\frac{x}{y} = (2\sqrt{2})$
On both sides of the above equation, taking the fourth power gives us,
$⇒\frac{x^{4}}{y^{4}} = (2 \sqrt{2})^{4}$
$⇒\frac{x^{4}}{y^{4}} = 4^{3}$
Hence, the correct answer is $4^3$.
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