Question

# The value of the expression 2(sin1∘+sin2∘+sin3∘+...+sin89∘)2(cos1∘+cos2∘+...+cos44∘)+1 is

A
2
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B
12
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C
12
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D
1
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Solution

## The correct option is B √2k=2(sin1+sin2+sin3+....+sin89)2(cos1+cos2+.....+cos44)+1⇒k=2cos45(sin(90−89)+sin(90−88)+sin(90−87)+.....+sin(90−89))(2cos1cos45)+(2cos2cos45)+....+(2cos44cos45)+cos45We know that 2cosAcosB=cos(A+B)+cos(A−B)∴k=√2(cos1+cos2+cos3+.....+cos89)(cos46+cos44)+(cos47+cos43)+....+(cos89+cos1)+cos45⇒k=√2(cos1+cos2+cos3+.....+cos89)(cos1+cos2+cos3+.....+cos89)⇒k=√2(cos1+cos2+cos3+.....+cos89)(cos1+cos2+cos3+.....+cos89)∴k=√2Ans: A

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