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Question

Mathematics Question on Inverse Trigonometric Functions

The value of tan⁡−113+tan⁡−115+tan⁡−117+tan⁡−118\tan^{-1}\frac{1}{3}+\tan^{-1}\frac{1}{5}+\tan^{-1}\frac{1}{7}+\tan^{-1}\frac{1}{8} is

A

π\pi

B

π4\frac{\pi}{4}

C

3π4\frac{3\pi}{4}

D

none of these.

Answer

π4\frac{\pi}{4}

Explanation

Solution

tan−113+tan−115+tan−117+tan−118tan^{-1} \frac{1}{3} +tan ^{-1} \frac{1}{5} +tan^{-1} \frac{1}{7} +tan^{-1} \frac{1}{8} =tan−113+151−115+tan−117+151−156= tan^{-1} \frac{\frac{1}{3}+\frac{1}{5}}{1-\frac{1}{15}} + tan^{-1}\frac{ \frac{1}{7}+\frac{1}{5}}{1- \frac{1}{56}} =tan−1814+tan−11555= tan^{-1} \frac{8}{14} + tan^{-1} \frac{15}{55} =tan−147+tan−1311= tan^{-1} \frac{4}{7} +tan^{-1} \frac{3}{11} =tan−147+3111−47⋅311+tan−16565= tan^{-1} \frac{\frac{4}{7}+\frac{3}{11}}{1-\frac{4}{7}\cdot\frac{3}{11}} + tan^{-1} \frac{65}{65} =tan−11=π4 = tan^{-1} 1 = \frac{\pi}{4}