If x, y, z are natural numbers such that $c o t^{- 1} x + c o t^{- 1} y = c o t^{- 1} z$ then the number of ordered triplets (x,y,z) that satisfy the equation is ___

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Infinite solutions

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Solution

The correct option is D Infinite solutions
x =1 : y =1 is not a solution of the given equation
Suppose $(x, y) \neq (1, 1)$
Then $\frac{\frac{1}{x} + \frac{1}{y}}{1 - \frac{1}{x y}} = \frac{1}{z} \Rightarrow x (z - y) = - (1 + y z) \Rightarrow= - \frac{(1 + y z)}{z - y}$
If y=n+1:z=n then $x = n^{2} + n + 1$
All such numbers are solutions

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