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Question
Prove the following:
sin8θ − cos8θ = (sin2θ − cos2θ) (1 − 2 sin2θ cos2θ)
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Solution
L.H.S. = sin8θ − cos8θ
= (sin4θ)2 – (cos4θ)2
= (sin4θ – cos4θ) (sin4θ + cos4θ)
= [(sin2θ)2 – (cos2θ)2] . [(sin2θ)2 + (cos2θ)2]
= (sin2θ + cos2θ) (sin2θ – cos2θ) . [(sin2θ + cos2θ)2 – 2sin2θ.cos2θ] …[∵ a2 + b2 = (a + b)2 – 2ab]
= (1) (sin2θ – cos2θ) (12 – 2sin2θ cos2θ)
= (sin2θ – cos2θ) (1 – 2sin2θ cos2θ)
= R.H.S.
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