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# Find the value of $\tan^{-1}1+\tan^{-1}2+\tan^{-1}3.$

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• $tan^{-1}1=\large\frac{\pi}{4}$
• $tan^{-1}x+tan^{-1}y=tan^{-1}\large\frac{x+y}{1-xy}+\pi$ if $xy > 1$
• $tan^{-1}(-1)=-\large\frac{\pi}{4}$
By taking x=2 and y=3, $\large\frac{x+y}{1-xy}=\large\frac{2+3}{1-6}=\large\frac{5}{-5}=-1$
$tan^{-1}2+tan^{-1}3=tan^{-1}\large\frac{2+3}{1-6}+\pi$
$=tan^{-1}(-1)+\pi=-\large\frac{\pi}{4}+\pi$
$and\: tan^{-1}1=\large\frac{\pi}{4}$

Substituting the values in the given expression we get
$tan^{-1}1+(tan^{-1}2+tan^{-1}3)=\large\frac{\pi}{4}+(-\large\frac{\pi}{4}+\pi)$
$=\pi$

answered Feb 20, 2013
edited Mar 19, 2013