If x 45^ = y cosec 30^ , then x^4y^4 is equal to?

Answer: Option A

$$\begin{array}{rl}& x\sin 45^{\circ }=y\mathrm{cosec}30^{\circ }\\ & \Rightarrow \frac{x}{y}=\frac{\text{cosec 3}{\text{0}}^{\circ }}{\text{sin }{45}^{\circ }}\\ & \Rightarrow \frac{x}{y}=\frac{2}{\frac{1}{\sqrt{2}}}\\ & \Rightarrow \frac{x}{y}=\frac{2\sqrt{2}}{1}\\ & \Rightarrow \frac{x^4}{y^4}={\left(\frac{2\sqrt{2}}{1}\right)}^4\\ & \Rightarrow \frac{x^4}{y^4}=\frac{64}{1}\\ & \Rightarrow \frac{x^4}{y^4}=4^3\end{array}$$