Let x,y and z be positive real numbers. Suppose x,y and z are lengths of the sides of a triangle  opposite to its angles X,Y and Z, respectively. If tan⁡X2+tan⁡Z2=2yx+y+z then which of the following statements is/are TRUE?

tan⁡x2+tan⁡z2=2yx+y+zΔS(S−x)+ΔS(S−z)=2y2SΔS2S−(x+z)(S−x)(S−z)=yS⇒ ΔyS(S−x)(S−z)=yS⇒ Δ2=(S−x)2(S−z)2⇒ S(S−y)=(S−x)(S−z)⇒ (x+y+z)(x+z−y)=(y+z−x)(x+y−z)⇒ (x+z)2−y2=y2−(z−x)2⇒ (x+z)2+(x−z)2=2y2⇒ x2+z2=y2⇒∠Y=π2⇒ ∠Y=∠X+∠Ztan⁡x2=ΔS(S−x)tan⁡x2=12xz(y+z)2−x24tan⁡x2=2xzy2+z2+2yz−x2tan⁡x2=2xz2z2+2yz  using y2=x2+z2tan⁡x2=xy+z