A geometry problem by Aakhyat Singh

Let $y = x(\frac{cos(A)}{cos(A+\frac{2\pi}{3})})$ , $z = x(\frac{cos(A)}{cos(A+\frac{4\pi}{3})})$ for $x,y,z \neq 0$ AND $A \neq (2n-1) \cdot \frac{\pi}{2} (n \in \mathbb{Z})$ . Thus:

$\frac{1}{x} + \frac{1}{y} + \frac{1}{z} = (\frac{1}{x}) \cdot [1 + \frac{cos(A+\frac{2\pi}{3})}{cos(A)} +\frac{cos(A+\frac{4\pi}{3})}{cos(A)} ];$

or $(\frac{1}{x}) \cdot [\frac{cos(A) + (cos(A)cos(\frac{2\pi}{3}) - sin(A)sin(\frac{2\pi}{3})) + (cos(A)cos(\frac{4\pi}{3}) - sin(A)sin(\frac{4\pi}{3})) }{cos(A)}];$

or $(\frac{1}{x}) \cdot [\frac{cos(A) + (cos(A)(-\frac{1}{2}) - sin(A)(\frac{\sqrt{3}}{2}) ) + (cos(A)(-\frac{1}{2}) - sin(A)(\frac{-\sqrt{3}}{2})) }{cos(A)}];$

or $(\frac{1}{x}) \cdot \frac{cos(A) - cos(A)}{cos(A)};$

or $(\frac{1}{x}) \cdot (1-1);$

or $\boxed{0}.$