Question

Evaluate: $\tan \left(100°\right)+\tan \left(125°\right)+\tan \left(100°\right)\tan \left(125°\right)$

• A. $0$
• B. $\frac{1}{2}$
• C. $-1$
• D. $1$

Answer 1

The correct option is D

$1$

Explanation for the correct option:

We know that,

$100°+125°=225°$

Take tan on both sides, we have

$\Rightarrow \tan \left(100°+125°\right)=\tan \left(225°\right)$

Using the formula, $\tan \left(a+b\right)=\frac{\tan \left(a\right)+\tan \left(b\right)}{1-\tan \left(a\right)\tan \left(b\right)}$

$$\begin{gathered} \Rightarrow \frac{\tan \left(100°\right)+\tan \left(125°\right)}{1-\tan \left(100°\right)\tan \left(125°\right)}=1\left[\because \tan \left(225°\right)=1\right] \\ \Rightarrow \tan \left(100°\right)+\tan \left(125°\right)=1-\tan \left(100°\right)\tan \left(125°\right) \\ \Rightarrow \tan \left(100°\right)+\tan \left(125°\right)+\tan \left(100°\right)\tan \left(125°\right)=1 \end{gathered}$$

Hence, the correct answer is option (D).