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

cosec⁴ θ - cosec² θ = cot⁴ θ + cot² θ

The L.H.S of the equation can be written as,

⇒ cosec² θ(cosec² θ - 1)
⇒ cosec² θ. cot² θ
⇒ (1 + cot² θ). cot² θ
⇒ cot² θ + cot⁴ θ.

Since, L.H.S. = R.H.S. hence, proved that cosec⁴ θ - cosec² θ = cot⁴ θ + cot² θ.