Trigonometric EquationHard
Question
3(sin x - cos x)4 + 6(sin x + cos x)2 + 4(sin6 x + cos6 x) equals to
Options
A.11
B.12
C.13
D.14
Solution
3(sin x - cos x)4 + 6(sin x + cos x)2 + 4(sin6 x + cos6 x)
= 3(sin x - cos x)4 + 6(sin x + cos x)2 + 4{(sin6 x + cos6 x) -3sin2 x cos2 x(sin2 x + cos2 x)}
= 3(1 - 2sin 2x + sin2 2x) + 6 + 6sin 2x + 4(1-3sin2 x cos2 x)
= 3(1 - 2sin 2x + sin2 2x + 2 + 2sin 2x + 4
=13 + 3sin2 2x - 3sin2 2x = 13
= 3(sin x - cos x)4 + 6(sin x + cos x)2 + 4{(sin6 x + cos6 x) -3sin2 x cos2 x(sin2 x + cos2 x)}
= 3(1 - 2sin 2x + sin2 2x) + 6 + 6sin 2x + 4(1-3sin2 x cos2 x)
= 3(1 - 2sin 2x + sin2 2x + 2 + 2sin 2x + 4
=13 + 3sin2 2x - 3sin2 2x = 13
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