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Question

Find the value of tan9tan27tan63+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 4
tan9°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°) [sinCsinD=2cosC+D2sinCD2]

=2.2cos36°sin18°cos36°sin18°=4

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