Prove that tan 70^° = tan 20^° + 2tan 50^° — JEE Mathematics

We know that $70^\circ - 20^\circ = 50^\circ$. Taking tangent on both sides:

$$\tan(70^\circ - 20^\circ) = \tan 50^\circ$$

$$\frac{\tan 70^\circ - \tan 20^\circ}{1 + \tan 70^\circ\tan 20^\circ} = \tan 50^\circ$$

Note that $\tan 70^\circ = \tan(90^\circ - 20^\circ) = \cot 20^\circ$.
Therefore, $\tan 70^\circ \tan 20^\circ = \cot 20^\circ \tan 20^\circ = 1$.
Substitute this into the denominator:

$$\frac{\tan 70^\circ - \tan 20^\circ}{1 + 1} = \tan 50^\circ$$

$$\frac{\tan 70^\circ - \tan 20^\circ}{2} = \tan 50^\circ$$

$$\tan 70^\circ - \tan 20^\circ = 2\tan 50^\circ \implies \tan 70^\circ = \tan 20^\circ + 2\tan 50^\circ$$

(Hence Proved)