Prove that the following identities :

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

Taking LHS

= sin⁸ θ – cos⁸ θ

= (sin⁴ θ)² – (cos⁴ θ)²

= (sin⁴ θ – cos⁴ θ)(sin⁴ θ+ cos⁴ θ )

[∵ (a² – b²) = (a + b) (a – b)]

= {(sin² θ)² – (cos² θ)²}{(sin² θ)² + (cos² θ)²}

= (sin² θ + cos² θ) (sin² θ – cos² θ) [(sin² θ + cos² θ) – 2 sin² θ cos² θ]

[ ∵ (a² + b²) = (a +b)² – 2ab]

= (1)[ sin² θ –cos² θ][(1) – 2 sin² θ cos² θ]

= (sin² θ – cos² θ)(1 – 2 sin² θ cos² θ)

=RHS

Hence Proved