A) \[\frac{\pi }{2}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) \[\frac{\pi }{6}\]
Correct Answer: C
Solution :
Given two angles are \[{{\tan }^{-1}}2\] and \[{{\tan }^{-1}}3.\] Let third angle be \[\theta ,\] then, \[{{\tan }^{-1}}2+{{\tan }^{-1}}3+\theta ={{180}^{o}}\] \[\Rightarrow \] \[{{\tan }^{-1}}\left( \frac{2+3}{1-2\times 3} \right)={{180}^{o}}-\theta \] \[\Rightarrow \] \[\frac{5}{-5}=\tan ({{180}^{o}}-\theta )=-\tan \theta \] \[\Rightarrow \] \[\tan \theta =1=\tan \frac{\pi }{4}\] \[\Rightarrow \] \[\theta =\frac{\pi }{4}\]
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