If cos3xsin2x=a1sinx+a2sin2x+⋯+ansinnx∀x∈R, where an≠0, then which of the following is/are correct ?
A
a5=18
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B
a2+a3=38
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C
a4=0
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D
n=5
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Solution
The correct option is Dn=5 cos3xsin2x=(cos2x)(cosxsin2x)=(1+cos2x2)(2sin2xcosx2) =14(1+cos2x)(sin3x+sinx) =14[sin3x+sinx+12(2sin3xcos2x)+12(2cos2xsinx)] =14[sin3x+sinx+12(sin5x+sinx)+12(sin3x−sinx)] =14[sinx+32sin3x+12sin5x]∴a1sinx+a2sin2x+⋯+ansinnx=14[sinx+32sin3x+12sin5x]