Angle Addition Identities Trignometry I'm working on a problem with angle addition, using the trig identities and seem to have gotten stuck. See screenshot for the work I have done in the red area. Any assistance is appreciated.
Your work up to the last step is fine.  (Yes, you could have said $\tan 45^\circ = 1$ but 
$\sqrt{2}/\sqrt{2}$ is surely OK.)

After your last step, I would rationalize the denominator as follows:
$$
\frac{6+\sqrt{12}}{6-\sqrt{12}} = \frac{6+\sqrt{12}}{6+\sqrt{12}} \cdot \frac{6+\sqrt{12}}{6-\sqrt{12}} = \frac{36+12\sqrt{12}+12}{6^2-12}= \frac{48+12\sqrt{12}}{24} = 2+\frac{\sqrt{12}}{2} = 2+\sqrt{3}
$$
Your work up to the last step is fine.  (Yes, you could have said $\tan 45^\circ = 1$ but 
$\sqrt{2}/\sqrt{2}$ is surely OK.)

After your last step, I would rationalize the denominator as follows:
$$
\frac{6+\sqrt{12}}{6-\sqrt{12}} = \frac{6+\sqrt{12}}{6+\sqrt{12}} \cdot \frac{6+\sqrt{12}}{6-\sqrt{12}} = \frac{36+12\sqrt{12}+12}{6^2-12}= \frac{48+12\sqrt{12}}{24} = 2+\frac{\sqrt{12}}{2} = 2+\sqrt{3}
$$