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

Toolbox:
• $$tan^{-1}1=\large\frac{\pi}{4}$$
• $$tan^{-1}x+tan^{-1}y=tan^{-1}\large\frac{x+y}{1-xy}+\pi$$ 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
$$tan^{-1}1+(tan^{-1}2+tan^{-1}3)=\large\frac{\pi}{4}+(-\large\frac{\pi}{4}+\pi)$$
$$=\pi$$

edited Mar 19, 2013