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

$\dfrac{\text{1 + tan A}}{\text{sin A}} + \dfrac{\text{1 + cot A}}{\text{cos A}} = \text{2(sec A + cosec A)}$

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:

cosec⁶ A - cot⁶ A = 3cot² A cosec² A + 1.

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.