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

$\dfrac{\text{sin }A}{1 + \text{cot }A} - \dfrac{\text{cos }A}{1 + \text{tan }A}$ = sin A - cos A

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

sec⁴ A - tan⁴ A = 1 + 2 tan² A

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

sec⁶ A - tan⁶ A = 1 + 3 tan² A + 3 tan⁴ A.

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

$\dfrac{\text{cot θ + cosec θ - 1}}{\text{cot θ - cosec θ + 1}} = \dfrac{\text{1 + cos θ}}{\text{sin θ}}$.