Question

If (x + y + z ) = xyz, then arctan(x) +arctan(y) +arctan (z) is what?​

Answer 1

Step-by-step explanation:

tan ^-1 x + tan^-1 y + tan^-1 z = pi.

Now tan^-1(x+y/1-xy) + tan^-1z = pi

So tan^-1(x+y/1-xy + z/1-zx+zy/1-xy) = pi

So tan^-1(x+y+z -xyz/Denominator)= pi

So x+y+z-xyz/Denominator = tan pi

we know tan pi =0

So x+y+Z = xyz (PROVED)

Answer 2

$answer$

x+y+z=xyz

x+y+z−xyz=0

(∀ 1−xy−yz−zx≠0)

$\frac{x+y+z−xyz}{1−xy−yz−zx} =0$

$\frac{tan( {tan}^{ - 1} x)+tan( { \tan }^{ - 1} y)+tan( { \tan }^{ - 1} z)−tan(tan^{ - 1}x)tan(tan^{ - 1}y)tan(tan^{ - 1}z)}{1−tan(tan^{ - 1}x)tan(tan^{ - 1}y)−tan(tan^{ - 1}y)tan(tan^{ - 1}z)−tan(tan^{ - 1}z)tan(tan^{ - 1}x)</p><p>} = 0</p><p>$

$\tan( \tan ^{ - 1} x+tan ^{ - 1} y+tan ^{ - 1} z)=tanπ</p><p>$

$tan ^{ - 1} x+tan^{ - 1} y+tan^{ - 1} z=π</p><p>$

arctan(x) +arctan(y) +arctan (z) is π