# 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

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