Simplify the trigonometric expression: ( (1 - cos(2x)) / sin(2x) ) * ( sin(2x) / (1 + cos(2x)) )

The given expression is:

$$\left(\frac{1-\cos2x}{\sin2x}\right) \left(\frac{\sin2x}{1+\cos2x}\right)$$

We can cancel out $\sin2x$ from the numerator and denominator:

$$\frac{1-\cos2x}{1+\cos2x}$$

We use the identities:

• $1 - \cos2x = 2\sin^2x$
• $1 + \cos2x = 2\cos^2x$ Substitute these into the expression:

$$\frac{2\sin^2x}{2\cos^2x}$$

Cancel out the 2s:

$$\frac{\sin^2x}{\cos^2x}$$

Recall that $\frac{\sin x}{\cos x} = \tan x$. Therefore, $\frac{\sin^2x}{\cos^2x} = \tan^2x$.

$$\tan^2x$$