扩展：由积分近似求和

Approximate the following sum using an integral: $$S = \sin 1^{\circ} + \sin 3^{\circ} + \sin 5^{\circ} + ... + \sin 89^{\circ}.$$

(This problem is inspired by Problem #13 on the 2015 AIME I.) Use an integral to approximate: $$P = \prod_{k=1}^{45} \csc^2 (2k-1)^{\circ}.$$