(1) 求证: $\frac{\sin 1^{\circ}}{\sin k^{\circ} \sin (k+1)^{\circ}}=\frac{\cos k^{\circ}}{\sin k^{\circ}}-\frac{\cos (k+1)^{\circ}}{\sin (k+1)^{\circ}}$;
(2) 求值: $\frac{1}{\cos 0^{\circ} \cos 1^{\circ}}+\frac{1}{\cos 1^{\circ} \cos 2^{\circ}}+\cdots+\frac{1}{\cos 44^{\circ} \cos 45^{\circ}}$.