Question

Evaluate :
$\tan 7^{\circ }\tan {23}^{\circ }\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }$

Answer 1

To evaluate,

$\tan 7^{\circ }\tan {23}^{\circ }\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }$

We know that,

$\tan (90-\theta )=\cot \theta$ and $\cot \theta =\frac{1}{\tan \theta }$

$\tan 7^{\circ }=\tan ({90}^{\circ }-{83}^{\circ })=\cot {83}^{\circ }$

$\tan {23}^{\circ }=\tan ({90}^{\circ }-{67}^{\circ })=\cot {67}^{\circ }$

$\therefore \tan 7^{\circ }\tan {23}^{\circ }\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }$

$=\cot {83}^{\circ }\cot {67}^{\circ }\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }$

$=\frac{1}{\tan {83}^{\circ }}\frac{1}{\tan {67}^{\circ }}\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }$

$=\frac{\tan {60}^{\circ }\tan {67}^{\circ }\tan {83}^{\circ }}{\tan {83}^{\circ }\tan {67}^{\circ }}$

$=\tan {60}^{\circ }$

$=\sqrt{3}$