Trignometry - 3

Geometry Level 3

If, tan ⁡ − 1 x + tan ⁡ − 1 y + tan ⁡ − 1 z = π 2 \tan^{-1}x + \tan^{-1}y + \tan^{-1}z =\dfrac{\pi}{2} and x y , y z , x z < 1 xy, yz, xz < 1 . Then find the value of x y + y z + x z xy + yz + xz


The answer is 1.

Kunal Joshi by Kunal Joshi , 24, India

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1 solution

Farhabi Mojib
Oct 16, 2014

We know,
t a n − 1 x + t a n − 1 y = t a n − 1 x + y 1 − x y tan^{-1} x + tan^{-1} y = tan^{-1} \frac{x + y}{1 - xy}
Now,
t a n − 1 x + t a n − 1 y + t a n − 1 z tan^{-1} x + tan^{-1} y + tan^{-1} z = t a n − 1 x + y 1 − x y + t a n − 1 z tan^{-1} \frac{x + y}{1 - xy} + tan^{-1} z
= t a n − 1 x + y 1 − x y + z 1 − x + y 1 − x y z = tan^{-1} \frac{\frac{x + y}{1 - xy} + z}{1 - \frac{x + y}{1 - xy} z}
Since,
x y xy <1 then,
1 − x + y 1 − x y z = 0 1 - \frac{x + y}{1 - xy} z = 0 [Because t a n π 2 tan \frac{π}{2} is undefined]
Then,
z x + z y = 1 − x y zx + zy = 1 - xy
So,
x y + y z + x z = 1. \boxed{xy + yz + xz = 1.}


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