<s>[INST] Find $x$ such that $\arctan(3/2)+\dots=\arctan x$ Find $x$ such that
$$\arctan(3/2) + \arctan(5/4) + \arctan(-5/2) + \arctan(-8/3) = \arctan x.$$
 [/INST] You may use the fact if $x > 0$, the argument of the complex number $x+iy$ is $\tan^{-1} \left(\frac{x}{y}\right)$. So if you compute the product
$$(2+3i)(4+5i)(2-5i)(3-8i) = 920 - 531i$$
you get that 
$$\tan^{-1}\left(\frac{3}{2}\right) + \tan^{-1}\left(\frac{5}{4}\right) + \tan^{-1}\left(-\frac{5}{2}\right) + \tan^{-1}\left(-\frac{8}{3}\right) \equiv \tan^{-1}\left(-\frac{531}{920}\right) \mod 2\pi$$
By computing both side on a calculator, you can check this actually an equality.
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