Question

The value of $\frac{\tan \left(70°\right)-\tan \left(20°\right)}{\tan \left(50°\right)}$ is

• A. $2$
• B. $1$
• C. $0$
• D. $3$

Answer 1

The correct option is A

$2$

The explanation of the correct option

The given trigonometric expression: $\frac{\tan \left(70°\right)-\tan \left(20°\right)}{\tan \left(50°\right)}$.

$$\begin{gathered} \tan \left(50°\right)=\tan \left(70°-20°\right) \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{1+\tan \left(70°\right)\tan \left(20°\right)}\left[\because \tan \left(A-B\right)=\frac{\tan \left(A\right)-\tan \left(B\right)}{1+\tan \left(A\right)\tan \left(B\right)}\right] \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{1+\tan \left(90°-20°\right)\tan \left(20°\right)} \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{1+\cot \left(20°\right)\tan \left(20°\right)}\left[\because \tan \left(90°-x\right)=\cot \left(x\right)\right] \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{1+\left[{\displaystyle \frac{1}{\tan \left(20°\right)}}\times \tan \left(20°\right)\right]}\left[\because \cot \left(x\right)=\frac{1}{\tan \left(x\right)}\right] \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{1+1} \\ \Rightarrow \tan \left(50°\right)=\frac{\tan \left(70°\right)-\tan \left(20°\right)}{2} \\ \Rightarrow \frac{\tan \left(70°\right)-\tan \left(20°\right)}{\tan \left(50°\right)}=2 \end{gathered}$$

Hence, option A is correct .