Question

$\frac{\tan \left(80°\right)-\tan \left(10°\right)}{\tan \left(70°\right)}$ is equal to:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is C

$2$

Explanation for correct option:

Solving the given trigonometric expression:

The given expression is: $\frac{\tan \left(80°\right)-\tan \left(10°\right)}{\tan \left(70°\right)}$

We know that: $80°-10°=70°$

Taking $\tan$ on both the sides, we get:

$\tan \left(80°-10°\right)=\tan \left(70°\right)$

Using the trigonometric identity $\mathbf{tan}\left(a-b\right)\mathbf{=}\frac{\mathbf{tan}\left(a\right)\mathbf{-}\mathbf{tan}\left(b\right)}{\mathbf{1}\mathbf{+}\mathbf{tan}\left(a\right)\mathbf{tan}\mathbf{\left(}\mathbf{b}\mathbf{\right)}}$ we get:

$$\begin{gathered} \frac{\tan \left(80°\right)-\tan (10°)}{1+\tan (80°)\tan (10°)}=\tan (70°) \\ \Rightarrow \frac{\tan \left(80°\right)-\tan (10°)}{\tan (70°)}=1+\tan (80°)\tan (10°) \\ \Rightarrow \frac{\tan \left(80°\right)-\tan (10°)}{\tan (70°)}=1+\tan (80°)\tan (90°-80°) \\ \Rightarrow \frac{\tan \left(80°\right)-\tan (10°)}{\tan (70°)}=1+\tan (80°)\cot \left(80°\right)\left[\because \tan (90°-a)=\cot \left(a\right)\right] \\ \Rightarrow \frac{\tan \left(80°\right)-\tan \left(10°\right)}{\tan \left(70°\right)}=1+1 \\ \Rightarrow \frac{\tan \left(80°\right)-\tan \left(10°\right)}{\tan \left(70°\right)}=2 \end{gathered}$$

Hence, the correct answer is option (C).