`"Let "e^(y)=(sqrt(1+alpha)+sqrt(1-alpha))/(sqrt(1+alpha)-sqrt(1-alpha)) and "tan"(x)/(2)=sqrt((1-alpha)/(1+alpha)),alpha in [-1, 0]cup(0,1]." Then "((dy)/(dx))_(x=(pi)/(3))=`

Options

A. `1//2`
B. 1
C. 2
D. `1//3`

Answer

2

Explanation

`e^(y)=(1+sqrt((1-alpha)/(1+alpha)))/(1-sqrt((1-alpha)/(1+alpha)))=(1+tan""(x)/(2))/(1-tan""(x)/(2))=tan ((pi)/(4)+(x)/(2))`
`therefore" "e^(y).y'=(1)/(2)sec^(2)((pi)/(4)+(x)/(2))" (Diff.w.r.t.x)"`
`=(1)/(2)[1+tan^(2)((pi)/(4)+(x)/(2))]`
`=(1)/(2)[1+e^(2y)]`
`"For "x,=(pi)/(3),e^(y)=tan((pi)/(4)+(pi)/(6))=2+sqrt(3)`
`rArr" "y'=(1)/(2(2+sqrt(3)))[1+(2+sqrt(3))^(2)]`
`=(1)/(2(2+sqrt(3)))[8+4sqrt(3)]`
`=2`