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Question: The value of tan<sup>–1</sup> \(\left( \frac { 1 } { 2 } \tan 2 \mathrm {~A} \right)\) + tan<sup>–1<...

The value of tan–1 (12tan⁡2 A)\left( \frac { 1 } { 2 } \tan 2 \mathrm {~A} \right) + tan–1(cotA) + tan–1(cot3A), for 0< A < π4\frac { \pi } { 4 } , is

A

tan–1(2)

B

tan–1(cotA)

C

4tan–1(1)

D

2 tan–1(2)

Answer

4tan–1(1)

Explanation

Solution

We have tan–1 (12tan⁡2 A)+tan⁡−1(cot⁡A)+tan⁡−1(cot⁡3 A)\left( \frac { 1 } { 2 } \tan 2 \mathrm {~A} \right) + \tan ^ { - 1 } ( \cot \mathrm { A } ) + \tan ^ { - 1 } \left( \cot ^ { 3 } \mathrm {~A} \right)

= tan–1 (12tan⁡2 A)+tan⁡−1(cot⁡A+cot⁡3 A1−cot⁡4 A)+π\left( \frac { 1 } { 2 } \tan 2 \mathrm {~A} \right) + \tan ^ { - 1 } \left( \frac { \cot \mathrm { A } + \cot ^ { 3 } \mathrm {~A} } { 1 - \cot ^ { 4 } \mathrm {~A} } \right) + \pi

(cot A > 1)

= tan–1 (tan⁡A1−tan⁡2 A)+π+tan⁡−1(cot⁡A1−cot⁡2 A)\left( \frac { \tan \mathrm { A } } { 1 - \tan ^ { 2 } \mathrm {~A} } \right) + \pi + \tan ^ { - 1 } \left( \frac { \cot \mathrm { A } } { 1 - \cot ^ { 2 } \mathrm {~A} } \right)

= tan⁡−1(tan⁡A1−tan⁡2 A)+π−tan⁡−1(tan⁡A1−tan⁡2 A)\tan ^ { - 1 } \left( \frac { \tan \mathrm { A } } { 1 - \tan ^ { 2 } \mathrm {~A} } \right) + \pi - \tan ^ { - 1 } \left( \frac { \tan \mathrm { A } } { 1 - \tan ^ { 2 } \mathrm {~A} } \right)

= p = 4. π4\frac { \pi } { 4 } = tan–1 (1).