If -π/2 < x < 0 and y = tan^-1√((1 - cos 2x)/(1 + cos 2x)), find dy/dx

Question

If \(-\cfrac{\pi}2\) < x < 0 and y = tan^-1\(\sqrt{\cfrac{1-cos\,2{\text x}}{1+cos\,2{\text x}}},\) find \(\cfrac{dy}{d\mathrm x} \).

Answer 1

When \(-\cfrac{\pi}2\) < x < 0, tan x is negative. So, square root of tan² x in this condition is –tan x.

So, y = tan^-1\(\sqrt{tan^2\text x}\)

=tan^-1(-tan x)

=-tan^-1(tan x)

=-x

And so