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

tanπ6+tan5π16+tan(8π+π16)+tan(8π+5π16) (tanπ16−cotπ16)+(tan5π16−cot5π16) =(sinπ16cosπ16−cosπ16sinπ16)+sin5π16cos5π16−cos5π16sin5π16 =sin2π16−cos2π16cosπ16sinπ16+sin25π16−cos25π16cos5π16sin5π16 =−[cosπ8].2sinπ8−cos5π8.2sin5π8 =−2cotπ8−2cot5π8 =−2cotπ8+2tanπ8 =−2[cotπ8−tanπ8] =−2[cosπ8sinπ8−sinπ8cosπ8] =−2[cosπ4]sinπ8.cosπ8 =−4cosπ4sinπ4 =−4 Abosulte value =4

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The absolute value of the expression $\tan \frac{π}{16} + \tan \frac{5 π}{16} + \tan \frac{9 π}{16} + \tan \frac{13 π}{16}$ is

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answer is 4.

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Detailed Solution

\tan \frac{π}{6} + \tan \frac{5 π}{16} + \tan (\frac{8 π + π}{16}) + \tan (\frac{8 π + 5 π}{16})

$(\tan \frac{π}{16} - \cot \frac{π}{16}) + (\tan \frac{5 π}{16} - \cot \frac{5 π}{16})$

$= (\frac{\sin \frac{π}{16}}{\cos \frac{π}{16}} - \frac{\cos \frac{π}{16}}{\sin \frac{π}{16}}) + \frac{\sin \frac{5 π}{16}}{\cos \frac{5 π}{16}} - \frac{\cos \frac{5 π}{16}}{\sin \frac{5 π}{16}}$

$= \frac{\sin^{2} \frac{π}{16} - \cos^{2} \frac{π}{16}}{\cos \frac{π}{16} \sin \frac{π}{16}} + \frac{\sin^{2} \frac{5 π}{16} - \cos^{2} \frac{5 π}{16}}{\cos \frac{5 π}{16} \sin \frac{5 π}{16}}$