Sin⁸θ – cos⁸θ = (1 – 2sin²θcos²θ)(sin²θ – cos²θ). Prove identity.

formula: - (i) sin²θ + cos²θ = 1

(ii) a² – b² = (a + b)(a – b)

taking L.H.S

Sin⁸θ – cos⁸θ = (sin⁴θ)² – (cos⁴θ)²

⇒ Sin⁸θ – cos⁸θ = (sin⁴θ + cos⁴θ)(sin⁴θ – cos⁴θ)

using formula (i)

⇒ Sin⁸θ – cos⁸θ = ((sin²θ)² + (cos²θ)²)((sin²θ)² – (cos²θ)²)

using formula (ii)

⇒ Sin⁸θ – cos⁸θ = ((sin²θ + cos²θ)² – 2sin²θcos²θ)) (sin²θ + cos²θ)(sin²θ – cos²θ)

⇒ Sin⁸θ – cos⁸θ = (1 – 2sin²θcos²θ)(sin²θ – cos²θ)

⇒ Sin⁸θ – cos⁸θ = (sin²θ – cos²θ)(1 – 2sin²θcos²θ)

L.H.S = R.H.S

Hence, Proved.