Given Z = (5+i)/(2+3i). To express this in the form x+yi, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2+3i is 2-3i.

From part (a), we have $Z = 1 - i$.
To express a complex number $x+yi$ in polar form $r(\cos\theta + i\sin\theta)$, we find the modulus $r = \sqrt{x^2+y^2}$ and the argument $\theta = \arctan\left(\frac{y}{x}\right)$ (adjusted for quadrant).
For $Z = 1 - i$, we have $x=1$ and $y=-1$.
Calculate the modulus $r$:

Calculate the argument $\theta$:
Since $x=1$ (positive) and $y=-1$ (negative), $Z$ lies in the fourth quadrant.
The reference angle $\alpha$ is given by $\tan\alpha = \left|\frac{y}{x}\right| = \left|\frac{-1}{1}\right| = 1$.
So, $\alpha = \frac{\pi}{4}$.
For a complex number in the fourth quadrant, $\theta = -\alpha$.

Now substitute $r$ and $\theta$ into the polar form $r(\cos\theta + i\sin\theta)$:

Using the trigonometric identities $\cos(-\theta) = \cos\theta$ and $\sin(-\theta) = -\sin\theta$:

From part (a), $Z = 1 - i$.
From part (b), $Z = \sqrt{2}\left(\cos\frac{\pi}{4} - i\sin\frac{\pi}{4}\right)$.
Equate the two expressions for $Z$:

For two complex numbers to be equal, their real parts must be equal and their imaginary parts must be equal.
Equating the real parts:

Divide by $\sqrt{2}$: