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Question

Trigonometric series of the form
sin(AB)cosAcosB+sin(BC)cosBcosC+sin(CD)cosCcosD
=tanAtanD
As we know that,
sin(AB)cosAcosB=tanAtanB
Based on the above given information, find sum of the series
sinxcos3x+sin3xcos9x+sin9xcos27x+ upto n terms

A
12(cot3nxcotx)
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B
12(tan3nxtanx)
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C
12(tan3xtan3n1x)
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D
12(cot3xcot3n1x)
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Solution

The correct option is B 12(tan3nxtanx)
Sn=sinxcos3x+sin3xcos9x+sin9xcos27x++sin(3n1)xcos(3n)x
=12[2sinxcosxcos3xcosx+2sin3xcos3xcos9xcos3x+2sin(3n1)xcos(3n1)xcos(3n)xcos(3n1)x]
=12[sin2xcos3xcosx+sin6xcos9xcos3x+sin2(3n1)xcos(3n)xcos(3n1)x]
=12[sin(3xx)cos3xcosx+sin(9x3x)xcos9xcos3x+sin(3nx3n1)xcos(3n)xcos(3n1)x]
=12[(tan3xtanx)+(tan9xtan3x)++(tan(3n)xtan(3n1)x)]
=12(tan(3n)xtanx)

Alternate solution:
Substitute n=2, x=0 and x=π4 in the options
and then verify it with the given series

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