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Question

The value of 13k=11sin(π4+(k1)π)6)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(23)
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Solution

The correct option is C 2(31)
The value of
13k=11sin(π4+(k1)π6)sin(π4+kπ6)13k=12sinπ6[sin{π4+kπ6(π4+(k1)π6)}sin(π4+(k1)π6))sin(π4+kπ6)]13k=12[cot(π4+(k1)π6)cot(π4+kπ6)]=2[{cotπ4cot(π4+π6)}+{cot(π4+π6)cot(π4+2π6)}++{cot(π4+12π6)cot(π4+13π6)}]=2[cotπ4cot(π4+13π6)]=2[1cot5π12]=2[1313+1]=2[1(23)]=2(31)

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