# 3cot 31° tan 15° cot 27° tan 75° cot 63° cot 59°

Question:

3cot 31° tan 15° cot 27° tan 75° cot 63° cot 59°

Solution:

$3 \cot 31^{\circ} \tan 15^{\circ} \cot 27^{\circ} \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ}$

$=3 \cot \left(90^{\circ}-59^{\circ}\right) \tan 15^{\circ} \cot \left(90^{\circ}-63^{\circ}\right) \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ}$

$=3 \tan 59^{\circ} \tan 15^{\circ} \tan 63^{\circ} \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ} \quad\left(\because \cot \left(90^{\circ}-\theta\right)=\tan \theta\right)$

$=3 \tan 59^{\circ} \tan \left(90^{\circ}-75^{\circ}\right) \tan 63^{\circ} \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ}$

$=3 \tan 59^{\circ} \cot 75^{\circ} \tan 63^{\circ} \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ} \quad\left(\because \tan \left(90^{\circ}-\theta\right)=\cot \theta\right)$

$=3\left(\tan 59^{\circ} \cot 59^{\circ}\right)\left(\tan 63^{\circ} \cot 63^{\circ}\right)\left(\cot 75^{\circ} \tan 75^{\circ}\right)$

$=3\left(\tan 59^{\circ} \frac{1}{\tan 59^{\circ}}\right)\left(\tan 63^{\circ} \frac{1}{\tan 63^{\circ}}\right)\left(\frac{1}{\tan 75^{\circ}} \tan 75^{\circ}\right) \quad\left(\because \cot \theta=\frac{1}{\tan \theta}\right)$

$=3$

Hence, $3 \cot 31^{\circ} \tan 15^{\circ} \cot 27^{\circ} \tan 75^{\circ} \cot 63^{\circ} \cot 59^{\circ}=3$.