if `cot^(-1)[sqrt(cosalpha)]-tan^(-1)[sqrt(cosalpha)]=x` then `sinx` is equal to

Options

A. `tan^(2).(alpha)/(2)`
B. `cot^(2).(alpha)/(2)`
C. `tan alpha`
D. `cot.(alpha)/(2)`

Answer

A

Explanation

`cot^(-1) (sqrt(cos alpha)) - tan^(-1) (sqrt(cos alpha)) = x`
or `tan^(-1) ((1)/(sqrt(cos alpha))) - tan^(-1) (sqrt(cos alpha)) = x`
or `tan^(-1).((1)/(sqrt(cosalpha)) - sqrt(cos alpha))/(1 + (1)/(sqrt(cos alpha)) sqrt(cos alpha)) = x`
or `tan^(-1).(1 - cos alpha)/(2sqrt(cos alpha)) = x`
or `tan x = (1 - cos alpha)/(2sqrt(cos alpha))`
or `cot x = (2 sqrt(cos alpha))/(1 - cos alpha)`
or `cosec x = sqrt(1 + (4 cos alpha)/((1 -cos alpha)^(2))) = (1 + cos alpha)/(1-cos alpha)`
or `sin x = (1- cos alpha)/(1 + cos alpha) = (2 sin^(2) (alpha//2))/(2 cos^(2) (alpha//2)) = "tan"^(2) (alpha)/(2)`