Question

$\tan {70}^0=a\tan {50}^0+\tan {20}^0$
Find a

Answer 1

$\tan {70}^0=\tan ({50}^0+{20}^0)=\frac{\tan {50}^0+\tan {20}^0}{1-\tan {50}^0\tan {20}^0}$ (i)
or $\tan {70}^0(1-\tan {50}^0\tan {20}^0)=\tan {50}^0+\tan {20}^0$
or $\tan {70}^0-\tan {50}^0\tan {20}^0\tan {70}^0=\tan {50}^0+\tan {20}^0$
or $\tan {70}^0=\tan {50}^0\tan {20}^0\tan {70}^0+\tan {50}^0+\tan {20}^0$
$=\tan ({90}^0-{20}^0)\tan {50}^0\tan {20}^0+\tan {50}^0+\tan {20}^0$
$=\cot {20}^0\tan {50}^0\tan {20}^0+\tan {50}^0+\tan {20}^0$
$=\tan {50}^0+\tan {50}^0+\tan {20}^0=2\tan {50}^0+\tan {20}^0$
Ans: 2