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Mathematics Question on Properties of Inverse Trigonometric Functions

The value of [tan⁡−1(12)+tan⁡−1(13)]\left[\tan^{-1} \left( \frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right)\right] is :

A

π2\frac{\pi}{2}

B

2π2\frac{2 \pi}{2}

C

π4\frac{\pi}{4}

D

π6\frac{\pi}{6}

Answer

π4\frac{\pi}{4}

Explanation

Solution

Consider tan⁡−1(12)+tan⁡−1(13)\tan^{-1} \left(\frac{1}{2}\right) + \tan^{-1} \left(\frac{1}{3}\right) =tan⁡−1[12+131−1213]=tan⁡−1[3+266−16]= \tan^{-1} \left[\frac{\frac{1}{2} + \frac{1}{3}}{1- \frac{1}{2} \frac{1}{3}}\right] = \tan^{-1} \left[\frac{\frac{3+2}{6}}{\frac{6-1}{6}}\right] =tan⁡−1(56×65)=tan⁡−1(1)= \tan^{-1} \left( \frac{5}{6} \times\frac{6}{5}\right) = \tan^{-1} \left(1\right) =tan⁡−1[tan⁡π4]=[π4]= \tan^{-1} \left[\tan \frac{\pi}{4}\right] = \left[ \frac{\pi}{4} \right]