The system of the equations                      x−ycosθ+zcos2θ=0                      −xcosθ+y−zcosθ=0                       xcos2θ−ycosθ+z=0 Has non trivial solution for θ equals

|1−cosθcos2θ−cosθ1−cosθcos2θ−cosθ1|=0 Applying  C1→C1−C3,  then⇒|2sin2θ−cosθcos2θ01−cosθ-2sin2θ−cosθ1|=0⇒           2sin2θ|1−cosθcos2θ01−cosθ-1−cosθ1|=0⇒           2sin2θ{1(1−cos2θ)−1(cos2θ−cos2θ)}=0⇒    2sin2θ{−2cos2θ+1+cos2θ}=0⇒    2sin20{0}=0  ⇒  0=0Which is independent of θ .