Question

The value of $\tan 6^o\tan {42}^o\tan {66}^o\tan {78}^o$ is?

• A. $1$
• B. $1/2$
• C. $1/4$
• D. $1/8$

Answer 1

Answer: (A)

$1$

$\tan 6^o\tan {42}^o\tan {66}^o\tan {78}^o$

$\Rightarrow$ Multiply and divide by $\left(\tan {54}^o\tan {18}^o\right)$

$\frac{\left(\tan 6^o\tan {42}^o\tan {48}^o\right)}{\tan \left({54}^o\right)\times \tan {18}^o}\times \left(\tan {54}^o\tan {18}^o\right)$

$\Rightarrow \frac{\left(\tan 6^o\tan {66}^o\tan {54}^o\right)\times \left(\tan {48}^o\tan {78}^o\tan {42}^o\right)}{\tan {54}^o\tan {18}^o}$

Using,

$\tan \theta \tan (60+\theta )\tan (60-\theta )=\tan 3\theta$

Here $\theta =6^o$ and $\theta ={18}^o$

$\therefore \frac{\tan 3\left(6^o\right)\times \tan 3\left({18}^o\right)}{\tan {54}^o\tan {18}^o}$

$\frac{\tan {18}^o\times \tan {54}^o}{\tan {54}^o\tan {18}^o}$

Ans $1$