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Prove the following:3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6 x + cos6 x) = 13 |
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Answer» (sin x – cos x) sin6 x + cos6 x = (sin2 x)3 + (cos2 x)3 = (sin2 x + cos2 x)3 – 3 sin2 x cos2 x (sin2 x + cos2 x) ….. [∵ a3 + b3 = (a + b)3 – 3ab(a + b)] = 13 – 3 sin2 x cos2 x (1) = 1 – 3 sin2 x cos2 x L.H.S. = 3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6 x + cos6 x) = 3(1 – 4 sin x cos x + 4 sin2 x cos2 x) + 6(1 + 2 sin x cos x) + 4(1 – 3 sin2 x cos2 x) = 3 – 12 sin x cos x + 12 sin2 x cos2 x + 6 + 12 sin x cos x + 4 – 12 sin2 x cos2 x = 13 = R.H.S. |
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