Question:
tan 10° tan 15° tan 75° tan 80° = ?
(a) $\sqrt{3}$
(b) 1
(c) $\frac{1}{\sqrt{3}}$
(d) –1
Solution:
$\tan 10^{\circ} \tan 15^{\circ} \tan 75^{\circ} \tan 80^{\circ}$
$=\tan \left(90^{\circ}-80^{\circ}\right) \tan 15^{\circ} \tan \left(90^{\circ}-15^{\circ}\right) \tan 80^{\circ}$
$=\cot 80^{\circ} \tan 15^{\circ} \cot 15^{\circ} \tan 80^{\circ} \quad\left(\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right)$
$=\left(\frac{1}{\tan 80^{\circ}} \tan 80^{\circ}\right)\left(\tan 15^{\circ} \frac{1}{\tan 15^{\circ}}\right) \quad\left(\because \cot \theta=\frac{1}{\tan \theta}\right)$
$=1$
Hence, the correct option is (b).