By using the properties of definite integrals,evaluate the integral $\begin{align*}\int\limits_0^\frac{\Large \pi}{\Large 2}\frac{\sin x-\cos x}{1+\sin x\cos x}\;dx \end{align*}$

Question

$\begin{array}{1 1}\large \frac{\pi}{2} \\ \large \frac{\pi}{4} \\ \pi \\ 0 \end{array} $