let t = tan x/2 then dx = ?

Question

let t = tan x/2 then dx = ?

Answer 1

t = tan x/2

Compute the derivative of both sides wrt x

Then dt/dx = (1/2)sec^2(x/2)

Solving for dx yields dx=2cos^2(x/2)dt

Answer 2

t = tanx/2; x/2 = tan^-1t; x = 2tan^-1t; dx = 2/(1 + ·t²)·dt

Answer 3

derivative of sinx = cosx

derivative of cosx = -sinx

tanx = sinx/cosx

derivative of sinx/cosx = (cosxcosx - cos(-cosx))cos^2(x)= 1/cos^2(x) = sec^2(x)

derivative of tanx = sec^2(x)

derivative of tan(x/2) = (1/2)sec^2(x/2)

use the chain rule

or maybe you meant tanx/2 as (tanx)/2

then

derivative of (tanx)/2 = (1/2)sec^(x)

I'd go with the 1st interpretation of the problem though

tanx/2 as tan(x/2)