The question presents the value of $\sin \theta = \frac{y}{x}$ and asks for the value of $\cos \theta$. This is based on a standard trigonometric identity.

Using the Pythagorean identity:

$\sin^2 \theta + \cos^2 \theta = 1$

Substitute $\sin \theta = \frac{y}{x}$ into the identity:

$\left(\frac{y}{x}\right)^2 + \cos^2 \theta = 1$

$\frac{y^2}{x^2} + \cos^2 \theta = 1$

Now, solve for $\cos^2 \theta$:

$\cos^2 \theta = 1 - \frac{y^2}{x^2}$

$\cos^2 \theta = \frac{x^2 - y^2}{x^2}$

Taking the square root of both sides:

$\cos \theta = \pm \frac{\sqrt{x^2 - y^2}}{x}$

Now let's check the options:

• Option (A): $\frac{\sqrt{x^2 - y^2}}{x}$
• Option (B): $\frac{y}{\sqrt{x^2 - y^2}}$
• Option (C): $\frac{x}{\sqrt{x^2 - y^2}}$
• Option (D): $\frac{\sqrt{x^2 - y^2}}{y}$

The correct answer is Option (A): $\frac{\sqrt{x^2 - y^2}}{x}$.

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