# Find the value of the expression in $tan^{-1} \bigg( tan \frac{3\pi}{4} \bigg)$

Toolbox:
• $$tan(\pi-x)=-tanx$$
• $$tan^{-1}(-1)=-\frac{\pi}{4}$$
Given $tan^{-1} \bigg( tan \frac{3\pi}{4} \bigg)$:
$$\large \frac{3\pi}{4}=\pi-\frac{\pi}{4}$$
The given expression becomes $$tan^{-1}tan \bigg[ \pi-\frac{\pi}{4} \bigg]$$
Given $$tan(\pi-x)=-tanx$$, this becomes $tan^{-1} (-tan \frac{\pi}{4})$
$$\Rightarrow\:tan^{-1}(-tan\frac{\pi}{4})=tan^{-1}(-1)=-\frac{\pi}{4}$$
edited Mar 14, 2013