Question:

<p><strong>Step 1:</strong> Recognize the structure of the given expression.</p><p>We have: $y = \frac{1 - \tan^2\left(\frac{\pi}{4} - x\right)}{1 + \tan^2\left(\frac{\pi}{4} - x\right)}$</p><p><strong>Step 2:</strong> Apply the double angle formula for cosine.</p><p>Recall that: $\cos(2\theta) = \frac{1 - \tan^2(\theta)}{1 + \tan^2(\theta)}$</p><p>Let $\theta = \frac{\pi}{4} - x$</p><p>Then: $y = \cos\left(2\left(\frac{\pi}{4} - x\right)\right) = \cos\left(\frac{\pi}{2} - 2x\right)$</p><p><strong>Step 3:</strong> Simplify using the cofunction identity.</p><p>$y = \cos\left(\frac{\pi}{2} - 2x\right) = \sin(2x)$</p><p>(Since $\cos\left(\frac{\pi}{2} - \alpha\right) = \sin(\alpha)$)</p><p><strong>Step 4:</strong> Find the maximum value.</p><p>The maximum value of $\sin(2x)$ is $\boxed{1}$, which occurs when $2x = \frac{\pi}{2} + 2k\pi$, i.e., $x = \frac{\pi}{4} + k\pi$ for integer $k$.</p><p><strong>∴ Answer:</strong> P</p>