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Question
If sinx + cosx = a, then sin6x + cos6x = ______.
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Solution
Given that: sinx + cosx = a
(sinx + cosx)2 = a2
⇒ sin2x + cos2x + 2sinx cosx = a2
⇒ 1 + 2sinx cosx = a2
⇒ sinx cosx = `(a^2 - 1)/2` .......(i)
sin6x + cos6x = (sin2x)3 + (cos2x)3
= (sin2x + cos2x)3 – 3 sin2x cos2x (sin2x + cos2x)
= `(1)^3 - 3((a^2 - 1)/2)^2 . 1`
= `1 - (3(a^2 - 1)^2)/4`
= `1/4[4 - 3(a^2 - 1)^2]`
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