Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

(sec A + tan A)(1 - sin A) = cos A.

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

(1 + tan² A)(1 - sin A)(1 + sin A) = 1.

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

(1 - cos A)(1 + sec A) = tan A sin A.

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

cot² A - cos² A = cot² A cos² A