Regarding Phase Angle Of a Fourier Transform
You just need to compute the phase of the denominator and add a minus sign because for any complex number \$z\$ you have

$$\arg\left\{\frac{1}{z}\right\}=-\arg\{z\}$$

where \$\arg\{\}\$ is the argument (or phase) of a complex number. So in your case

$$z=1-ae^{-jw}=1-a(\cos w-j\sin w)=1-a\cos w+ja\sin w$$

So the phase of \$X(e^{j w})\$ is

$$\phi_X(w)=-\arctan\left(\frac{a\sin w}{1-a\cos w}\right)\pm\pi$$

where the possible addition (or subtraction) of \$\pi\$ is determined by the signs of the numerator and the denominator inside the \$\arctan()\$ function. In many programming languages the computation of the correct phase value is implemented as 'atan2()', so you don't need to worry about adding or subtracting \$\pi\$ after evaluating the \$\arctan()\$ function.