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 (

So the equation

*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*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 = n*π

for any integer

*n*.
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