If π < x < 3π/2, Then Find the Value of √[(1 − cos 2x)/(1 + cos 2x)]

Question

If \[ \pi

Solution

Using the identities

\[ 1-\cos2x=2\sin^2x \] and \[ 1+\cos2x=2\cos^2x \]

Therefore,

\[ \sqrt{\frac{1-\cos2x}{1+\cos2x}} = \sqrt{\frac{2\sin^2x}{2\cos^2x}} \] \[ = \sqrt{\tan^2x} = |\tan x| \]

Since

\[ \pi \(x\) lies in the third quadrant where \(\tan x\) is positive. Hence,

\[ |\tan x| = \tan x \]

Therefore,

\[ \sqrt{\frac{1-\cos2x}{1+\cos2x}} = \tan x \]

Answer

\[ \boxed{\tan x} \]