<s>[INST] On the proof $\tan 70°-\tan 20° -2 \tan 40°=4\tan 10°$ I am currently studying in class 10 and I am unable to do this problem. $$\tan 70 ° -\tan 20° -2 \tan 40° =4\tan 10°$$ Can anybody please help me.
Thanks!
 [/INST] \begin{align*}
 \tan 70-\tan 20 & =\tan 70-\cot 70\\
 & =\tan 70-\frac{1}{\tan 70}\\
 & = \frac{\tan^2 70-1}{\tan 70}=-\frac{1-\tan^2 70}{\tan 70}\\
 & = -\frac{2(1-\tan^2 70)}{2\tan 70}=-\frac{2}{\frac{2\tan 70}{1-\tan^2 70}}\\
 & = \frac{-2}{\tan 140}= \frac{-2}{-\tan 40}= \frac{2}{\tan 40}=2\cot 40 
 \end{align*}
    Now, $$ \tan 70-\tt-2\tan 40 =2\cot 40-2\tan 40=-2(\tan 40-\cot 40)  $$
    \begin{align*}
 -2(\tan 40-\cot 40)= -2\left( \tan 40-\frac{1}{\tan 40} \right) 
 \end{align*}
    Now do the same as above to get the result. You will get $ 4\cot 80=4\tan 10. $
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