Question

Prove that $\tan 50°-\tan 40°=2\tan 10°$

Answer 1

To Prove

$\tan 50°-\tan 40°=2\tan 10°$

Use formula:

$\tan (a+b)=\frac{(\tan a+\tan b)}{(1-\tan a\tan b)}$

Consider

$$\begin{gathered} \tan 50=\tan (40+10) \\ \Rightarrow \tan 50=\frac{(\tan 40+\tan 10)}{(1-\tan 40\tan 10)}Crossmultiply \\ \Rightarrow \tan 50(1-\tan 40\tan 10)=\tan 40+\tan 10 \\ \Rightarrow \tan 50–\tan 50\tan 40\tan 10=\tan 40+\tan 10\because \tan 50=\tan (90-40)=cot40andcot40\tan 40=1 \\ \Rightarrow \tan 50–\tan 10=\tan 40+\tan 10 \\ \Rightarrow \tan 50–\tan 40=\tan 10+\tan 10 \\ \Rightarrow \tan 50–\tan 40=2\tan 10 \end{gathered}$$

Hence, proved.