Question Number 126381 by liberty last updated on 20/Dec/20
$$\:\mathrm{arctan}\:\mathrm{1}\:+\:\mathrm{arctan}\:\mathrm{2}\:+\:\mathrm{arctan}\:\mathrm{3}\:=? \\ $$
Answered by benjo_mathlover last updated on 20/Dec/20
$$\left(\bullet\right)\:\mathrm{arctan}\:\mathrm{1}+\mathrm{arctan}\:\mathrm{2}=\mathrm{arctan}\:\left(\frac{\mathrm{1}+\mathrm{2}}{\mathrm{1}−\mathrm{1}.\mathrm{2}}\right) \\ $$$$\:\mathrm{arctan}\:\left(−\mathrm{3}\right) \\ $$$$\left(\bullet\bullet\right)\:\mathrm{arctan}\:\left(−\mathrm{3}\right)+\mathrm{arctan}\:\left(\mathrm{3}\right)=\mathrm{arctan}\:\left(\frac{−\mathrm{3}+\mathrm{3}}{\mathrm{1}−\left(−\mathrm{3}\right)\left(\mathrm{3}\right)}\right) \\ $$$$\:=\:\mathrm{arctan}\:\left(\frac{\mathrm{0}}{\mathrm{7}}\right)\:=\:\mathrm{arctan}\:\mathrm{0}\:=\:\pi \\ $$
Commented by talminator2856791 last updated on 20/Dec/20
$$\:\mathrm{brilliant} \\ $$
Commented by mathmax by abdo last updated on 20/Dec/20
$$\mathrm{arctan}\left(\mathrm{0}\right)=\mathrm{0} \\ $$
Commented by liberty last updated on 20/Dec/20
$${correct}...{answer}\:{is}\:\pi \\ $$
Answered by mathmax by abdo last updated on 20/Dec/20
$$\mathrm{arctan}\left(\mathrm{1}\right)=\frac{\pi}{\mathrm{4}}\:\mathrm{and}\:\mathrm{tan}\left(\mathrm{arctan2}+\mathrm{arctan3}\right)=\frac{\mathrm{2}+\mathrm{3}}{\mathrm{1}−\mathrm{2}.\mathrm{3}}=\frac{\mathrm{5}}{−\mathrm{5}}=−\mathrm{1} \\ $$$$\Rightarrow\mathrm{arctan2}\:+\mathrm{arctan3}=\mathrm{arctan}\left(−\mathrm{1}\right)=−\frac{\pi}{\mathrm{4}}\:\Rightarrow\Sigma\mathrm{arctan}=\mathrm{0} \\ $$