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

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

2 sec² θ - sec⁴ θ - 2 cosec² θ + cosec⁴ θ = cot⁴ θ - tan⁴ θ.

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

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