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

sin² θ + cos⁴ θ = cos² θ + sin⁴ θ.

Solving L.H.S.,

⇒ sin² θ + cos⁴ θ

= 1 - cos² θ + (cos² θ)²

= 1 - cos² θ + (1 - sin² θ)²

= 1 - cos² θ + 1 + sin⁴ θ - 2sin² θ

= 1 - cos² θ + 1 + sin⁴ θ - 2(1 - cos² θ)

= 2 - cos² θ + sin⁴ θ - 2 + 2cos² θ

= 2cos² θ - cos² θ + sin⁴ θ

= cos² θ + sin⁴ θ.

Since, L.H.S. = R.H.S. hence, proved that sin² θ + cos⁴ θ = cos² θ + sin⁴ θ.