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Question

The absolute value of the expression tanπ16+tan5π16+tan9π16+tan13π16 is

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Solution

tanπ16+tan5π16+tan9π16+tan13π16sinπ16cosπ16+sin9π16cos9π16+sin5π16cos5π16+sin13π16cos13π16

weknowthatsin(A+B)=sinAcosB+cosAsinBAnd2cosAcosB=cos(A+B)+cos(AB)

Sowegetsinπ16cos9π16+sin9π16cosπ16cosπ16cos9π16+sin5π16cos13π16+sin13π16cos5π16cos5π16cos13π162sin10π16cos10π16+cosπ2+2sin18π16cos18π16+cosπ22sin5π8cos5π8+2sin9π8cos9π82sin14π8cos5π8cos9π84sin7π4cos7π4+cosπ24tan7π4=4Sotheabsolutevalueis4

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