Prove that: sin6 x + cos6 x = 1 - 3 sin2 x cos2 x

Let us consider

LHS: sin6 x + cos6 x (sin2 x) 3 + (cos2 x) 3

By using the formula, a3 + b3 = (a + b) (a2 + b2 – ab)

(sin2 x + cos2 x) [(sin2 x) 2 + (cos2 x) 2 – sin2 x cos2 x]

By 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

Hence proved.

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