I came across an article lately, which has continued to amaze me. The simple equation
X + Y + Z = XYZ
has many solutions such as (X,Y,Z) = (1,2,3) or (0,√3,-√3,). But to find ALL real solutions, trigonometry surprisingly gives a perfect answer. All we need is the identity
Consider any solution (X,Y,Z). Since the range of the tangent function is the set of all real numbers, we can always find real numbers A, B and C such that
So the equation
X = tan A, Y = tan B and Z = tan C.
So the equation
X + Y + Z = XYZ
is equivalent to
tan A + tan B + tan C = tan A tan B tan C,
which is equivalent to
tan (A + B + C) = 0,
yielding the solutions
A + B + C = nπ
for any integer n.
