Prove the following with the help of identities:

sin⁸θ – cos⁸θ = (sin²θ – cos²θ) (1– 2sin²θ cos²θ)

Taking L.H.S we get,

Sin⁸ θ– cos⁸ θ

Using the formula: a² – b² = (a – b)(a + b)

⇒ (sin⁴ θ – cos⁴ θ)(sin⁴ θ + cos⁴ θ)

⇒ (sin² θ – cos² θ)(sin² θ + cos² θ)((sin² θ + cos² θ)² – 2.sin² θ.cos² θ)

⇒ (sin² θ – cos² θ)(1)((1)² – 2.sin² θ.cos² θ) (Using: sin² θ + cos² θ = 1.)

⇒ (sin² θ – cos² θ)(1 – 2.sin² θ.cos² θ)

= R.H.S

Hence, proved