Question

tanx + tan 4x + tan 7x = tanxtan4xtan7x

solve for x with detailed explainations

Answer 1

$tan(a + b + c) = \frac{tan(a) + tan(b) + tan(c) - tan(a) tan(b) tan(c)}{1 - (tan(a)tan(b) + tan(b)tan(c) + tan(c)tan(a)}$

Now consider a + b + c = nπ
then tan(a + b + c) = 0

So, tan(a) + tan(b) + tan(c) - tan(a)tan(b)tan(c) = 0
-> tan(a) + tan(b) + tan(c) = tan(a)tan(b)tan(c)

Therefore, x + 4x + 7x = nπ
-> 12x = nπ

-> x = $\frac{n \pi }{12}$