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Question

# Find the value of tan9∘−tan27∘−tan63∘+tan81∘.

A
4
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B
3
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C
2
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D
1
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Solution

## The correct option is D 4tan9°−tan27°−tan63°+tan81°=tan9°+tan81°−(tan27°+tan63°)=tan9°+tan(90°−9°)−(tan27°+tan(90°−27°))=tan9°+cot9°−(tan27°+tan27°) [∵cotA=tan(90˚−A)]=sin9°cos9°+cos9°sin9°−(sin27°cos27°+cos27°sin27°)=sin29°+cos29°sin9°cos9°−(sin227°+cos227°sin27°cos27°)=1sin9°cos9°−1sin27°cos27°=2sin18°−2sin54° [∵sin2x=2sinxcosx]=2[sin54°−sin18°sin18°sin54°]=2.2cos36°sin18°sin18°sin(90°−36°) [∵sinC−sinD=2cosC+D2sinC−D2]=2.2cos36°sin18°cos36°sin18°=4

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