`[a^3-3a+(a^2-1)sqrt[a^2-4]-2]/[a^3-3a+(a^2-1)sqrt[a^2-4]+2] (a>2)`

`= [(a^3-3a-2)+(a^2-1)sqrt[a^2-4]]/[(a^3-3a+2)+(a^2-1)sqrt[a^2-4]]`

`= [[(a^3-4a)+(a-2)]+(a^2-1)sqrt[a^2-4]]/[[(a^3-4a)+(a+2)]+(a^2-1)sqrt[a^2-4]]`

`= [[a(a-2)(a+2)+(a-2)]+(a^2-1)sqrt[a^2-4]]/[[a(a-2)(a+2)+(a+2)]+(a^2-1)sqrt[a^2-4]]`

`= [(a-2)(a^2+2a+1)+(a-1)(a+1)sqrt[a^2-4]]/[(a+2)(a^2-2a+1)+(a-1)(a+1)sqrt[a^2-4]]`

`= [(a-2)(a+1)^2+(a-1)(a+1)sqrt[a^2-4]]/[(a+2)(a-1)^2+(a-1)(a+1)sqrt[a^2-4]]`

`= [(a+1)[(a-2)(a+1)+(a-1)sqrt[a^2-4]]]/[(a-1)[(a-1)(a+2)+(a+1)sqrt[a^2-4]]]`

`= [(a+1){sqrt[a-2][sqrt[a-2](a+1)+(a-1)sqrt[a+2]]}]/[(a-1){sqrt[a+2][sqrt[a+2](a-1)+(a+1)sqrt[a-2]]}]`

`=[(a+1)sqrt[a-2]]/[(a-1)sqrt[a+2]]`

ʙʟᴀᴄᴋꜱᴛᴀʀᴠɴツ