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Question: If cot<sup>–1</sup>α + cot<sup>–1</sup>β = cot<sup>–1</sup> x, then x =...

If cot–1α + cot–1β = cot–1 x, then x =

A

α + β

B

α – β

C

1+αβα+β\frac{1 + \alpha\beta}{\alpha + \beta}

D

αβ−1α+β\frac{\alpha\beta - 1}{\alpha + \beta}

Answer

αβ−1α+β\frac{\alpha\beta - 1}{\alpha + \beta}

Explanation

Solution

cot–1 α + cot–1 β = cot–1 x

tan–1 (1α)\left( \frac { 1 } { \alpha } \right)+ tan–1 (1β)\left( \frac { 1 } { \beta } \right)= tan–1

⇒ tan–1 [1α+1β1−1α⋅1β]\left[ \frac { \frac { 1 } { \alpha } + \frac { 1 } { \beta } } { 1 - \frac { 1 } { \alpha } \cdot \frac { 1 } { \beta } } \right]= tan–1

⇒ β+ααβαβ−1αβ\frac { \frac { \beta + \alpha } { \alpha \beta } } { \frac { \alpha \beta - 1 } { \alpha \beta } }= ⇒ x = αβ−1α+β\frac { \alpha \beta - 1 } { \alpha + \beta }