Question

Solve $\tan {20}^o\tan {40}^o\tan {80}^o$ is equal to

• A. $\frac{1}{\sqrt{3}}$
• B. $\sqrt{3}$
• C. $3$
• D. $1$

Answer 1

Answer: (B)

$\sqrt{3}$

$\tan {20}^o\tan {40}^o\tan {80}^o$

$\Rightarrow$ $\frac{2\sin \left(10\right)\cos \left(10\right).2\sin \left(20\right)\cos \left(20\right).2\sin \left(40\right)\cos \left(40\right)}{\cos \left(20\right).\cos \left(40\right).\sin \left(10\right)}$

$\Rightarrow$ $2\cos \left(10\right).2\sin \left(20\right).2\sin \left(40\right)$

$\Rightarrow$ $2.2\cos \left(10\right).\left[2\sin \right(40\left)\sin \right(20\left)\right]$

$\Rightarrow$ $2.2\cos \left(10\right).\left[\cos \right(40-20)-\cos (40+20\left)\right]$

$\Rightarrow$ $2.2\cos \left(10\right).\left[\cos \right(20)-\cos (60\left)\right]$

$\Rightarrow$ $2.2\cos \left(10\right).\left[2{\cos }^2\right(10)-1-\frac{1}{2}]$

$\Rightarrow$ $2.2\cos \left(10\right).\left[2{\cos }^2\right(10)-\frac{3}{2}]$

$\Rightarrow$ $2.\left[4{\cos }^3\right(10)-\frac{3}{2}.2\cos (10\left)\right]$

$\Rightarrow$ $2.\left[4{\cos }^3\right(10)-2\cos (10\left)\right]$

$\Rightarrow$ $2\cos (3\times 10)$

$\Rightarrow$ $2\cos \left(30\right)$

$\Rightarrow$ $2\times \frac{\sqrt{3}}{2}$

$\Rightarrow$ $\sqrt{3}$