If `u=cot^-1 sqrt(tanalpha)-tan^-1 sqrt(tan alpha),` then `tan(pi/4-u/2)` is equal to (a) `sqrt(tan alpha)` (b) `sqrt(cos alpha)` (c) `tan alpha` (d)

Monjur Alahi

If `u=cot^-1 sqrt(tanalpha)-tan^-1 sqrt(tan alpha),` then `tan(pi/4-u/2)` is equal to (a) `sqrt(tan alpha)` (b) `sqrt(cos alpha)` (c) `tan alpha` (d) `cot alpha`
A. `sqrt(tan alpha)`
B. `sqrt(cot alpha)`
C. `tan alpha`
D. `cot alpha`


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Salim Ahmed Kawsar

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Correct Answer - A
Let `sqrt(tan alpha) = tan x`. Then
`u = cot^(-1) (tan x) - tan^(-1) (tan x)`
`= (pi)/(2) - x -x = (pi)/(2) - 2x`
or `2x = (pi)/(2) -u`
or `x = (pi)/(4) - (u)/(2)`
or `tan x = tan ((pi)/(4) -(u)/(2))`
`rArr sqrt(tan alpha) = tan ((pi)/(4) - (u)/(2))`

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