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

$\dfrac{\text{cos A}}{\text{cosec A + 1}} + \dfrac{\text{cos A}}{\text{cosec A - 1}} = \text{2 tan A}$.

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

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

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

$\dfrac{\text{cot θ}}{\text{cosec θ + 1}} + \dfrac{\text{cosec θ + 1}}{\text{cot θ}} = \text{2 sec θ}.$

Prove the following trigonometry identity :

(sin θ + cos θ)(cosec θ - sec θ) = cosec θ.sec θ - 2 tan θ