Let $I = \int\limits_{0}^{\pi/2} \frac{\sin x - \cos x}{1 + \sin x \cos x} dx$ ...(i) Then $\int\limits_{0}^{\pi/2} \frac{\sin(\pi/2 - x) - \cos(\pi/2 - x)}{1 + \sin(\pi/2 - x) \cos(\pi/2 - x)} dx$ $\left[\because \int\limits_{0}^{a} f(x) dx = \int\limits_{0}^{a} f(a-x) dx\right]$ $I = \int\limits_{0}^{\pi/2} \frac{\cos x - \sin x}{1 + \cos x \sin x} dx$ ...(ii) Adding (i) and (ii), we get $2I = \int\limits_{0}^{\pi/2} \frac{\sin x - \cos x + \cos x - \sin x}{1 + \cos x \sin x} dx$ $2I = \int\limits_{0}^{\pi/2} 0 dx$ $\Rightarrow I = 0$