If `xsin45^(@)=y"cosec " 30^(@)`, then `(x^(4))/(y^(4))` is equal to

Options

A. `4^(3)`
B. `6^(3)`
C. `2^(3)`
D. `8^(3)`

Answer

To solve the equation \( x \sin 45^\circ = y \csc 30^\circ \) and find the value of \( \frac{x^4}{y^4} \), we can follow these steps:

### Step 1: Substitute the values of \( \sin 45^\circ \) and \( \csc 30^\circ \)

We know that:
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \csc 30^\circ = \frac{1}{\sin 30^\circ} = \frac{1}{\frac{1}{2}} = 2 \)

So, we can rewrite the equation as:
\[ x \cdot \frac{1}{\sqrt{2}} = y \cdot 2 \]

### Step 2: Rearrange the equation

Now, we can rearrange this equation to express \( \frac{x}{y} \):
\[ x = y \cdot 2 \sqrt{2} \]

### Step 3: Find \( \frac{x}{y} \)

From the equation \( x = 2\sqrt{2} \cdot y \), we can express \( \frac{x}{y} \):
\[ \frac{x}{y} = 2\sqrt{2} \]

### Step 4: Raise both sides to the power of 4

Now, we need to find \( \frac{x^4}{y^4} \):
\[ \frac{x^4}{y^4} = \left(\frac{x}{y}\right)^4 = (2\sqrt{2})^4 \]

### Step 5: Calculate \( (2\sqrt{2})^4 \)

Calculating \( (2\sqrt{2})^4 \):
\[ (2\sqrt{2})^4 = 2^4 \cdot (\sqrt{2})^4 = 16 \cdot 4 = 64 \]

### Final Answer

Thus, the value of \( \frac{x^4}{y^4} \) is:
\[ \frac{x^4}{y^4} = 64 \]

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