If \[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) \] prove that \[ xy+yz+zx=0 \]
Solution
Let
\[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) =k \]Then
\[ x=\frac{k}{\cos\theta} \] \[ y=\frac{k}{\cos\left(\theta+\frac{2\pi}{3}\right)} \] \[ z=\frac{k}{\cos\left(\theta+\frac{4\pi}{3}\right)} \]Now,
\[ xy+yz+zx \] \[ = k^2\left[ \frac1{\cos\theta\cos\left(\theta+\frac{2\pi}{3}\right)} \right. \] \[ + \frac1{ \cos\left(\theta+\frac{2\pi}{3}\right) \cos\left(\theta+\frac{4\pi}{3}\right) } \] \[ \left. + \frac1{ \cos\left(\theta+\frac{4\pi}{3}\right)\cos\theta } \right] \]Take L.C.M.:
\[ = \frac{ k^2[ \cos\left(\theta+\frac{4\pi}{3}\right) +\cos\theta +\cos\left(\theta+\frac{2\pi}{3}\right) ] }{ \cos\theta \cos\left(\theta+\frac{2\pi}{3}\right) \cos\left(\theta+\frac{4\pi}{3}\right) } \]Use identity:
\[ \cos\theta + \cos\left(\theta+\frac{2\pi}{3}\right) + \cos\left(\theta+\frac{4\pi}{3}\right) =0 \]
\[
xy+yz+zx=0
\]
Hence Proved.