If x/cos θ = y/cos(θ − 2π/3) = z/cos(θ + 2π/3), Find x + y + z

Solution

Let

\[ \frac{x}{\cos\theta} = \frac{y}{\cos\left(\theta-\frac{2\pi}{3}\right)} = \frac{z}{\cos\left(\theta+\frac{2\pi}{3}\right)} =k \]

Then,

\[ x=k\cos\theta \]

\[ y=k\cos\left(\theta-\frac{2\pi}{3}\right) \]

\[ z=k\cos\left(\theta+\frac{2\pi}{3}\right) \]

Therefore,

\[ x+y+z = k\left[ \cos\theta + \cos\left(\theta-\frac{2\pi}{3}\right) + \cos\left(\theta+\frac{2\pi}{3}\right) \right] \]

Using the identities:

\[ \cos\left(\theta-\frac{2\pi}{3}\right) = -\frac{1}{2}\cos\theta+\frac{\sqrt3}{2}\sin\theta \]

and

\[ \cos\left(\theta+\frac{2\pi}{3}\right) = -\frac{1}{2}\cos\theta-\frac{\sqrt3}{2}\sin\theta \]

Adding all three terms,

\[ \cos\theta -\frac{1}{2}\cos\theta -\frac{1}{2}\cos\theta + \frac{\sqrt3}{2}\sin\theta -\frac{\sqrt3}{2}\sin\theta =0 \]

Hence,

\[ x+y+z=0 \]

Therefore,

\[ \boxed{0} \]