Question

tan20° + 2 tan50° =​

Answer 1

SOLUTION:-

Given:

tan20° + 2tan50°

Proof:

We know that, 20 +50= 70

Taking tan on both sides, we get;

tan(20+50)= tan70.

$\frac{(tan20 + tan50)}{(1 - tan20.tan50)} = tan70 \: \: \: [∵tan \:(a + b) = \frac{(tana + tanb)}{(1 - tan \: a.tanb)} )] \\ \\ = > tan20 + tan50 = tan70 \times (1 - tan20.tan50) \\ \\ = > tan20 + tan50 = tan70 - tan20.tan50.tan70 \\ \\ = > tan20 + tan50 = tan70 - tan20tan50cot20 \: \: \: \: [tan70 = tan(90 - 20) = cot20] \\ \\ = > tan20 + tan50 = tan70 - tan20.tan50.( \frac{1}{tan20} ) \: \: \: \:[∵cot20 = \frac{1}{tan20} ]\\ \\ = > tan20 + tan50 = tan70 - tan50 \\ \\ = > tan20 + tan50 + tan50 = tan70 \\ \\ = > 2tan5 0 + tan20 = tan70 \\ \\ = > 2tan50 + tan20 = cot20 \: \: \: \:[∵tan70 = cot20]$

Proved.

Hope it helps ☺️