Question

Prove that: tan 5 tan 35 tan 60 tan55 tan 85 = root 3

Answer 1

$tan\,5^{\circ}{\times}tan\,35^{\circ}{\times}tan\,60^{\circ}{\times}tan\,55^{\circ}{\times}tan\,85^{\circ}$

$\text{Using,}$

$\boxed{\bf\;tan\,\theta=cot(90^{\circ}-\theta)}$

$\implies\;tan5^{\circ}=cot\,85^{\circ}\;\text{and}$

$tan35^{\circ}=cot\,55^{\circ}$

$\text{Now,}$

$tan\,5^{\circ}{\times}tan\,35^{\circ}{\times}tan\,60^{\circ}{\times}tan\,55^{\circ}{\times}tan\,85^{\circ}$

$=cot\,85^{\circ}{\times}cot\,55^{\circ}{\times}\sqrt{3}{\times}tan\,55^{\circ}{\times}tan\,85^{\circ}$

$=\frac{1}{tan\,85^{\circ}}{\times}\frac{1}{tan\,55^{\circ}}{\times}\sqrt{3}{\times}tan\,55^{\circ}{\times}tan\,85^{\circ}$

$=\sqrt{3}$

$\therefore\boxed{\bf\,tan\,5^{\circ}{\times}tan\,35^{\circ}{\times}tan\,60^{\circ}{\times}tan\,55^{\circ}{\times}tan\,85^{\circ}=\sqrt{3}}$