Use the tangent subtraction formula.

To prove the identity $\tan^2\left(\frac{1}{4}\pi - \frac{1}{2}\theta\right) = \frac{1 - \sin\theta}{1 + \cos\theta}$, we will start with the left-hand side (LHS) and transform it into the right-hand side (RHS).

Step 1: Use the tangent subtraction formula.
The formula for $\tan(A - B)$ is $\frac{\tan A - \tan B}{1 + \tan A \tan B}$.
Let $A = \frac{\pi}{4}$ and $B = \frac{\theta}{2}$.

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