Trig2.6

Prove that $\tan 50 =\tan 40 +2\tan 10 \,$

We have 40+10=50

Now $\tan (40+10)=\tan 50\,$

$\frac{\tan 40+\tan 10}{1-\tan 40 \tan 20}=\tan 50\,$

Cross multiplying,we get

$\tan 40+\tan 10=\tan 50-\tan 50 \tan 40 \tan 10\,$

$\tan 40+\tan 10=\tan 50-\tan (90-40) \tan 40 \tan 10\,$

$\tan 40+\tan 10=\tan 50-\cot 50 \tan 40 \tan 10\,$

$\tan 40+\tan 10=\tan 50- \tan 10\,$ (Since tan and cot are inverse to eachother)

$\tan 40+\tan 10+\tan 10=\tan 50\,$

Therefore, $\tan 40+2\tan 10=\tan 50\,$