Show that:
$tan 70^{o} = tan 20^{o} + 2 tan 50^{o}$ .

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Solution

$t a n 70^{\circ} = t a n 20^{\circ} + 2 t a n 50^{\circ}$
$70^{\circ} = 20 + 50$
$t a n 70 = t a n (20 + 50)$
$t a n 70 = \frac{t a n 20 + t a n 50}{1 - t a n 20 t a n 50}$
$t a n 70 - t a n 70 t a n 20 t a n 50 = t a n 20 + t a n 50$
$t a n 70 - t a n 70 t a n (90 - 70) t a n 50 = t a n 20 + t a n 50$
$t a n 70 - t a n 70 c o t 70 t a n 50 = t a n 20 + t a n 50$
$t a n 70 - t a n 50 = t a n 20 + t a n 50 (∴ t a n x w t x = 1)$
$t a n 70 = t a n 20 + 2 t a n 50$

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tan 70^{o} - tan 20^{o} = 2 tan 50^{o}

tan 70^{o} - tan 20^{o} = 2 tan 50^{o}

tan 20^{o} + 2 tan 50^{o}

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2 tan 10^{o} + tan 50^{o} = 2 x, tan 20^{0} + tan 50^{o} = 2 y, 2 tan 10^{o} + tan 70^{o} = 2 w

tan 20^{o} + tan 70^{o} = 2 z

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