Question

prove that $\tan 70°=\tan 20°+2\tan 50°$

Answer 1

To prove: $\tan 70°=\tan 20°+2\tan 50°$

Take:

$$\begin{gathered} \tan 70=\tan (20+50)\because \mathit{t}\mathit{a}\mathit{n}\mathbf{}\mathbf{(}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{B}\mathbf{)}=\frac{(\tan A+\tan B)}{1–\tan A\tan B} \\ \Rightarrow \tan 70=\frac{(\tan 20+\tan 50)}{1–\tan 20\tan 50} \\ \Rightarrow \tan 70(1–\tan 20\tan 50)=\tan 20+\tan 50 \\ \Rightarrow \tan 70–\tan 70\tan 20\tan 50=\tan 20+\tan 50 \\ \Rightarrow \tan 70-\tan 50=\tan 20+\tan 50(\because \tan 70\tan 20=1) \\ \Rightarrow \tan 70=\tan 20+2\tan 50 \end{gathered}$$

Hence proved.