prove that $\displaystyle tan50^{o} -tan40^{o} = 2tan10^{o} $
Hi Tweety,
I don't know what formulas you have been taught, but here's another proof using the Tan sum formula:
$\displaystyle \tan(u+v)=\frac{\tan u+\tan v}{1-\tan u \tan v}$.
$\displaystyle \tan (50) = \tan (40+10)= \frac{\tan 40 + \tan 10}{1- \tan 40 \tan 10}$
$\displaystyle \tan 50 (1- \tan 40 \tan 10) = \tan 40 + \tan 10
$
Now, distribute:
$\displaystyle \tan 50 - {\color{red}\tan 50 \tan 40} \tan 10 = \tan 40 + \tan 10$
You know that $\displaystyle {\color{red}\tan 50 \tan 40 = 1}$
$\displaystyle \tan 50 - \tan 10= \tan 40 + \tan 10$
$\displaystyle \tan 50 = \tan 40 + 2 \tan 10$
$\displaystyle \tan 50 - \tan 40 = 2 \tan 10$