Prove each of the following identities:

Consider L.H.S. =

Using identities (a³ + b³) = (a + b)(a² + b² – ab) and (a³ – b³) = (a – b)(a² + b² + ab)

∴ L.H.S. =

= (cos² θ + sin² θ – cos θ sin θ) + (cos² θ + sin² θ + cos θ sin θ)

= (1 – cos θ sin θ) + (1 + cos θ sin θ)

= 2

= R.H.S.

Hence, proved.