# Prove identity

• Jul 29th 2009, 10:54 PM
arze
Prove identity
Prove that $\displaystyle \arctan x+\arctan y\equiv \arctan\frac{x+y}{1-xy}$.
use this relationship to show that, if $\displaystyle \arctan x+\arctan y+arctan z=\frac{\pi}{2}$ then $\displaystyle xy+yz+xz=1$
I have done the first part:
Let $\displaystyle \arctan x=\theta$ and $\displaystyle \arctan y=\phi$
then for LHS
$\displaystyle \equiv \theta +\phi$
$\displaystyle \equiv \arctan(\tan(\theta+\phi))$
$\displaystyle \equiv \arctan(\frac{tan \theta+\tan \phi}{1-\tan \theta \tan \phi})$
$\displaystyle \equiv \arctan(\frac{x+y}{1-xy})\equiv RHS$
now my problem is the second part, after i subsitute $\displaystyle \arctan(\frac{x+y}{1-xy})$ into $\displaystyle \arctan x+\arctan y+arctan z=\frac{\pi}{2}$ to get $\displaystyle \arctan(\frac{x+y}{1-xy})+\arctan z=\frac{\pi}{2}$, i don't know what to do.
Thanks
• Jul 29th 2009, 10:58 PM
red_dog
Let $\displaystyle \frac{x+y}{1-xy}=w$

Apply again the first identity for $\displaystyle \arctan w+\arctan z$ and replace back w.
• Jul 29th 2009, 11:21 PM
arze
thanks! i did that, and let $\displaystyle arctan w=\omega$ and $\displaystyle arctan z=\gamma$
so i had $\displaystyle \omega+\gamma=\frac{\pi}{2}$
$\displaystyle arctan(tan(\omega+\gamma)=\frac{\pi}{2}$
$\displaystyle arctan(\frac{w+z}{1-wz})=\frac{\pi}{2}$
$\displaystyle arctan(\frac{\frac{x+y}{1-xy}+z}{1-z(\frac{x+y}{1-xy})})=\frac{\pi}{2}$
$\displaystyle arctan(\frac{x+y+z-xyz}{1-xy-xz-yz})=\frac{\pi}{2}$
now i'm stuck, $\displaystyle tan\frac{\pi}{2}$ doesn't give a value
• Jul 29th 2009, 11:29 PM
red_dog
If $\displaystyle \arctan\frac{a}{b}=\frac{\pi}{2}$ then b must be 0.

In this case $\displaystyle 1-xy-xz-yz=0\Rightarrow xy+xz+yz=1$

Edited.
• Jul 29th 2009, 11:31 PM
arze
i thinking something along that line but didn't have any confirmation
thanks!
• Jul 30th 2009, 12:28 AM
Quote:

Originally Posted by red_dog
If $\displaystyle \arctan\frac{a}{b}=0$ then b must be 0.

In this case $\displaystyle 1-xy-xz-yz=0\Rightarrow xy+xz+yz=1$

Whoops! I think you mean 'If $\displaystyle \arctan\frac{a}{b}=\frac{\pi}{2}$ then b must be 0'.

• Jul 30th 2009, 12:45 AM
red_dog
Quote:

Whoops! I think you mean 'If $\displaystyle \arctan\frac{a}{b}=\frac{\pi}{2}$ then b must be 0'.