Question:

Given the equation $\tan\left(\frac{X}{2}\right) + \tan\left(\frac{Z}{2}\right) = \frac{2y}{x + y + z}$, assume $x, y, z$ are sides of a triangle with semi-perimeter $s = \frac{x + y + z}{2}$. Using the half-angle tangent identity for a triangle: $$ \tan\left(\frac{A}{2}\right) = \sqrt{\frac{(s - b)(s - c)}{s(s - a)}} $$ Apply this to angles $X$ and $Z$ (opposite sides $x$ and $z$): $$ \tan\left(\frac{X}{2}\right) + \tan\left(\frac{Z}{2}\right) = \sqrt{\frac{(s - y)(s - z)}{s(s - x)}} + \sqrt{\frac{(s - x)(s - y)}{s(s - z)}} $$ Factor out $\sqrt{\frac{s - y}{s}}$ and simplify the remaining terms using algebraic manipulation. After simplification, equate the result to $\frac{2y}{x + y + z}$ and verify consistency with triangle identities.