Find the value of $\frac{\cos 25^{\circ }-\sin 65^{\circ }}{\cos 25^{\circ }+\sin 65^{\circ }}$.

Using the complementary angle identity $\sin (90^{\circ }-\theta )=\cos \theta$:
$$\sin 65^{\circ }=\sin (90^{\circ }-25^{\circ })=\cos 25^{\circ }$$
Substitute $\sin 65^{\circ }=\cos 25^{\circ }$ into the given expression:
$$\frac{\cos 25^{\circ }-\cos 25^{\circ }}{\cos 25^{\circ }+\cos 25^{\circ }}=\frac{0}{2\cos 25^{\circ }}=0$$