Prove that $tan 80^{o} = tan 10^{o} + 2 tan 70^{o}$

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Solution

$80 = 10 + 70$
Taking tangent of both sides,

$tan 80 = tan (10 + 70)$
$\Rightarrow tan 80$ = $\frac{t a n 10 + t a n 70}{1 - t a n 10. t a n 70}$

$\Rightarrow t a n 80 - t a n 80. t a n 10. t a n 70 = t a n 10 + t a n 70$ (Since, tan80.tan10 = cot10.tan10 = 1)
$\Rightarrow t a n 80 - t a n 70 = t a n 10 + t a n 70$
$\Rightarrow t a n 80 = t a n 10 + 2 t a n 70$
Hence Proved.

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