Question

Show that $\tan {36}^{\circ }\tan {17}^{\circ }\tan {54}^{\circ }\tan {73}^{\circ }=1$

Answer 1

To prove that:

$\tan {36}^o\tan {17}^o\tan {54}^o\tan {73}^o=1$

Proof:

$\tan \theta =\cot ({90}^o-\theta )$

$\tan \theta =\frac{1}{\tan ({90}^o-\theta )}$

Hence, $\tan {54}^o=\frac{1}{\tan ({90}^o-{54}^o)}$

$=\frac{1}{\tan {36}^o}$

$\tan {54}^o\tan {36}^o=1...............\left(i\right)$

Similarly, $\tan {17}^o\tan {73}^o=1.............\left(ii\right)$

Multiplying (i) and (ii), we get

$\tan {54}^o\tan {36}^o\tan {17}^o\tan {73}^o=1$

Hence, proved