# Find the exact value of the expression. ( tan25^(circ) +tan110^(circ))/(1-tan25^(circ)*tan110^(circ))

Find the exact value of the expression.
$\mathrm{tan}{25}^{\circ }+\mathrm{tan}{110}^{\circ }/1-\mathrm{tan}{25}^{\circ }\cdot \mathrm{tan}{110}^{\circ }$

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Macsen Nixon

40
use the identity:
$\mathrm{tan}\left(u+v\right)=\left(\mathrm{tan}u+\mathrm{tan}v\right)/\left(1-\mathrm{tan}u\cdot \mathrm{tan}v\right)$
where:
$u={25}^{\circ },v={110}^{\circ }$
Therefore,
$\left(\mathrm{tan}{25}^{\circ }+\mathrm{tan}{110}^{\circ }\right)/\left(1+\mathrm{tan}{25}^{\circ }\cdot \mathrm{tan}{110}^{\circ }\right)=\mathrm{tan}\left({25}^{\circ }+{110}^{\circ }\right)$
$\left(\mathrm{tan}{25}^{\circ }+\mathrm{tan}{110}^{\circ }\right)/\left(1+\mathrm{tan}{25}^{\circ }\cdot \mathrm{tan}{110}^{\circ }\right)=\left(\mathrm{tan}{135}^{\circ }\right)$
$\left(\mathrm{tan}{25}^{\circ }+\mathrm{tan}{110}^{\circ }\right)/\left(1+\mathrm{tan}{25}^{\circ }\cdot \mathrm{tan}{110}^{\circ }\right)$
Result: $-1$