$tan 20^{o} + tan 25^{o} + tan 20^{o} tan 25^{o}$ .

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Solution

As, we know, $t a n 45^{\circ} = 1.$
take $45^{\circ} = 20^{\circ} + 25^{\circ}$
So, $t a n (20^{\circ} + 25^{\circ}) = 1$
$\frac{t a n 20^{\circ} + t a n 25^{\circ}}{1 - t a n 20^{\circ} t a n 25^{\circ}} = 1$
So, $t a n 20^{\circ} + t a n 25^{\circ} = 1 - t a n 20^{\circ} t a n 25^{\circ}$
$t a n 20^{\circ} + t a n 25^{\circ} + t a n 20^{\circ} + t a n 20^{\circ} t a n 25^{\circ} = 1$
So, we get, $t a n 20^{\circ} + t a n 25^{\circ} + t a n 20^{\circ} t a n 25^{\circ}$
$= 1.$

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