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Question
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
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Solution
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = 13.
Explanation:
Given expression is 3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x)
= 3[sin2x + cos2x – 2 sinx cosx]2 + 6(sin2x + cos2x + 2sinx cosx) + 4[(sin2x)3 + (cos2x)3]
= 3[1 – 2sinx cosx]2 + 6(1 + 2sinx cosx) + 4[(sin2x + cos2x)3 – 3sin2x cos2x (sin2x + cos2x)]
= 3[1 + 4sin2x cos2x – 4sinx cosx] + 6(1 + 2 sinx cosx) + 4[1 – 3sin2x cos2x]
= 3 + 12sin2x cos2x – 12sinx cosx + 6 + 12sinx cosx + 4 – 12sin2x cos2x
= 3 + 6 + 4
= 13
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