Someone can explain me why $\tan(-\frac{\pi}4+\arctan x)=\frac{x-1}{x+1}$

Question

Someone can explain me why $tan(-\frac{\pi}4+\arctan x)=\frac{x-1}{x+1}$??

I try to understand it, bot I don't understand how to came from one side to the other...

Thank you!

Answer 1

We know that $$\tan(a-b)=\dfrac{\tan a-\tan b}{1+\tan a\tan b}$$ So, let $b=\pi/4,a=\arctan x$. Then, $$\tan(\arctan x-\pi/4)=\dfrac{\tan \arctan x-\tan \pi/4}{1+\tan \arctan x\tan \pi/4}=\dfrac{x-1}{1+x}$$