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Mathematics

Prove the following identities, where the angles involved are acute angles for which the trigonometric ratios are defined:

2(sin6 θ + cos6 θ) - 3(sin4 θ + cos4 θ) + 1 = 0.

Trigonometric Identities

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Answer

Solving L.H.S.,

⇒ 2[(sin2 θ)3 + (cos2 θ)3] - 3[(sin2 θ)2 + (cos2 θ)2] + 1

⇒ 2[(sin2 θ + cos2 θ)3 - 3sin2 θ cos2 θ(sin2 θ + cos2 θ)] - 3[(sin2 θ + cos2 θ)2 - 2sin2 θ cos2 θ)] + 1

⇒ 2[(1)3 - 3sin2 θ cos2 θ(1)] - 3[(1)2 - 2sin2 θ cos2 θ)] + 1

⇒ 2 - 6sin2 θ cos2 θ - 3 + 6sin2 θ cos2 θ + 1

⇒ 2 - 3 + 1 - 6sin2 θ cos2 θ + 6sin2 θ cos2 θ

⇒ 3 - 3

⇒ 0.

Since, L.H.S. = R.H.S. hence, proved that 2(sin6 θ + cos6 θ) - 3(sin4 θ + cos4 θ) + 1 = 0.

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