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  4. If z+(1/z)=2 cos θ, z ∈ C then z2 n-2 zn cos (n θ) is equal to

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Q. If $z+\frac{1}{z}=2 \cos \theta, z \in C$ then $z^{2 n}-2 z^n \cos (n \theta)$ is equal to

Complex Numbers and Quadratic Equations

Solution:

$z+\frac{1}{z}=2 \cos \theta \Rightarrow z^2-2 z \cos \theta+1=0 $
$\Rightarrow z=\cos \theta \pm i \sin \theta $
$\text { Now, } z^{2 n}-2 z^n \cos (n \theta) $
$=z^n\left[z^n-2 \cos (n \theta)\right] $
$=z^n[\cos (n \theta) \pm i \sin (n \theta)-2 \cos (n \theta)] $
$=-z^n \bar{z}^n=-1$