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Prove the identity: 1 + cos2 2x = 2(cos4 x + sin4 x) |
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Answer» Let us consider the LHS 1 + cos2 2x As we know, cos2x = cos2 x – sin2 x cos2 x + sin2 x = 1 Therefore, 1 + cos2 2x = (cos2 x + sin2 x)2 + (cos2 x – sin2 x)2 = (cos4 x + sin4 x + 2 cos2 x sin2 x) + (cos4 x + sin4 x – 2 cos2 x sin2 x) = cos4 x + sin4 x + cos4 x + sin4 x = 2 cos4 x + 2 sin4 x = 2(cos4 x + sin4 x) = RHS Thus proved. |
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