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Question

The value of 13k=11sin(π4+(k1)π6)sin(π4+kπ6) is equal to

A
33
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B
2(33)
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C
2(31)
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D
2(2+3)
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Solution

The correct option is C 2(31)
13k=11sin(π4+(k1)π6)sin(π4+kπ6)

=213k=1sin[(π4+kπ6)(π4+(k1)π6)]sin(π4+(k1)π6)sin(π4+kπ6)

=213k=1sin(π4+kπ6).cos(π4+(k1)π6)sin(π4+(k1)π6)cos(π4+kπ6)sin(π4+(k1)π6)sin(π4+kπ6)
=213k=1[cot(π4+(k1)π6)cot(π4+kπ6)]

=2[cotπ4cot(π4+13π6)]=2[cotπ4cot(2π+π4+π6)]
=2[1cot(π4+π6)]=2⎢ ⎢ ⎢ ⎢11131+13⎥ ⎥ ⎥ ⎥=2[1313+1]=2×23+1=2(31)

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