Question

The value of $\tan \left(40°\right)+\tan \left(20°\right)+\sqrt{3}\tan \left(20°\right)\tan \left(40°\right)$ is equal to

• A. $\sqrt{12}$
• B. $\frac{1}{\sqrt{3}}$
• C. $1$
• D. $\frac{\sqrt{3}}{2}$
• E. $\sqrt{3}$

Answer 1

The correct option is E

$\sqrt{3}$

The explanation of the correct option

The given trigonometric expression: $\tan \left(40°\right)+\tan \left(20°\right)+\sqrt{3}\tan \left(20°\right)\tan \left(40°\right)$.

$$\begin{gathered} \tan \left(60°\right)=\tan \left(40°+20°\right) \\ \Rightarrow \sqrt{3}=\frac{\tan \left(40°\right)+\tan \left(20°\right)}{1-\tan \left(40°\right)\tan \left(20°\right)}\left[\because \tan \left(A+B\right)=\frac{\tan \left(A\right)+\tan \left(B\right)}{1-\tan \left(A\right)\tan \left(B\right)}\right] \\ \Rightarrow \sqrt{3}\left[1-\tan \left(40°\right)\tan \left(20°\right)\right]=\tan \left(40°\right)+\tan \left(20°\right) \\ \Rightarrow \tan \left(40°\right)+\tan \left(20°\right)+\sqrt{3}\tan \left(20°\right)\tan \left(40°\right)=\sqrt{3} \end{gathered}$$

Hence, option E is correct.