Mathematics - Properties of Triangle Question with Solution | TestHub

$$\begin{array}{rlrlr}& \frac{a+c}{\sqrt{a^2-ac+c^2}}=\frac{\sin A+\sin C}{\sqrt{{\sin }^{2}A-\sin A\sin C+{\sin }^{2}C}}=\frac{2\sin \frac{A+C}{2}\cos \frac{A-C}{2}}{\sqrt{\frac{1-\cos 2A}{2}-\frac{\cos (A-C)-\cos (A+C)}{2}+\frac{1-\cos 2C}{2}}}& =\frac{2\sqrt{2},\frac{\sqrt{3}}{2}\cos \frac{A-C}{2}}{\sqrt{2-(\cos 2A+\cos 2C)-\cos (A-C)+\cos (A+C)}}& =\frac{\sqrt{6}\cos \frac{A-C}{2}}{\sqrt{\frac{3}{2}-2\cos (A+C)\cos (A-C)-\cos (A-C)}}=\frac{\sqrt{6}\cos \frac{A-C}{2}}{\sqrt{\frac{3}{2}+\cos (A-C)-\cos (A-C)}}& =2\cos \frac{A-C}{2}\end{array}$$