Deductive proof and identities

An equation holds for its solutions; an identity holds for every value in its stated domain. The symbol can emphasise an identity. A proof links known facts to the required conclusion; numerical tests alone do not prove a universal claim. **Task** | **Start** | **Check** Prove an identity | Simplify one side until it becomes the other. | Keep every denominator nonzero. Prove a numerical fact | Use exact arithmetic. | Avoid rounded approximations in an exact proof. Check a result | Substitute into the original expression or reverse a valid calculation. | A successful check is evidence for that case, not a general proof. Example For x!= -1, prove x2-1x+1 x-1. LHS= (x-1)(x+1)x+1 =x-1 =RHS. The restriction x!= -1 is essential: cancelling a factor does not make the original fraction defined at -1.