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

$\dfrac{\text{cot A - 1}}{2 - \text{sec}^2 A} = \dfrac{\text{cot A}}{\text{1 + tan A}}$.

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

$\dfrac{1}{1 + \text{sin } θ} + \dfrac{1}{1 - \text{sin } θ}$ = 2 sec² θ

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

$\dfrac{\text{cos θ}}{\text{1 - tan θ}} - \dfrac{\text{sin}^2 θ}{\text{cos θ - sin θ}} = \text{cos θ + sin θ}$.

Prove that:

$\dfrac{(1 + \text{sin }θ)^2 + (1 - \text{sin }θ)^2}{2\text{ cos }^2θ}$ = sec²θ + tan²θ