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:

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

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

(cosec A - sin A)(sec A - cos A)sec² A = tan A.