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Prove the identity: sin6 x + cos6 x = 1 – 3 sin2 x cos2 x |
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Answer» Let us consider the LHS: sin6 x + cos6 x (sin2 x)3 + (cos2 x)3 On using the formula, a3 + b3 = (a + b) (a2 + b2 – ab) (sin2 x + cos2 x) [(sin2 x)2 + (cos2 x)2 – sin2 x cos2 x] On using the formula, sin2 x + cos2 x = 1 and a2 + b2 = (a + b)2 - 2ab 1 × [(sin2 x + cos2 x)2 – 2sin2 x cos2 x – sin2 x cos2 x 12 – 3sin2 x cos2 x 1 – 3sin2 x cos2 x = RHS ∴ LHS = RHS Thus proved. |
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