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Question

If cos3xsin2x=a1sinx+a2sin2x++ansinnx xR, where an0, 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 D n=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(sin3xsinx)]
=14[sinx+32sin3x+12sin5x]a1sinx+a2sin2x++ansinnx=14[sinx+32sin3x+12sin5x]

So,
n=5a5=18a2+a3=0+38=38a4=0

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