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

$\dfrac{\text{sec A - 1}}{\text{sec A + 1}} = \dfrac{\text{1 - cos A}}{\text{1 + cos A}}$

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

$\dfrac{\text{tan}^2 θ}{\text{(sec θ - 1)}^2} = \dfrac{\text{1 + cos θ}}{\text{1 - cos θ}}$

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

sec² A + cosec² A = sec² A cosec² A.

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

$\dfrac{\text{1 + sin A}}{\text{cos A}} + \dfrac{\text{cos A}}{\text{1 + sin A}} = 2\text{ sec A}$