Answer
$$x=a\sec\theta\\y=b\sin\theta$$
Work Step by Step
Draw a line such that the angle between the line and the $x$-axis is $\theta$. Draw two line segments perpendicular to the $x$-axis with each going through points where the line we drew earlier intersects with each circle.
The distance P is away from the $x$-axis is the height of the triangle inside the smaller circle with hypotenuse $b$, so we can say:
$$y=b\sin\theta$$
The length of the segment $OB$ is the distance point $P$ is from the $y$-axis, so let $OB=x$.
Since segment $AB$ is tangent to the outer circle, we know $\angle AOB$ is a right angle. Using the definition of trigonometric ratios, we can say:
$$\frac{a}{x}=\cos\theta\therefore\\x=a\sec\theta$$
