If z2-2 z cos θ+1=0, then z2+z-2 is equal to

Question

If z2-2 z cos θ+1=0, then z2+z-2 is equal to (A) 2 cos 2 θ (B) 2 sin 2 θ (C) 2 cos θ (D) 2 sin θ

Tardigrade Admin · Accepted Answer

We have, z2-2 z cos θ+1=0 ⇒ z=(2 cos θ ± √4 cos 2 θ-4/2) = cos θ ± √ cos 2 θ-1 = cos θ ± √- sin 2 θ = cos θ ± √i2 sin 2 θ = cos θ ± i sin θ When z= cos θ+i sin θ z2+z-2= cos 2 θ+i sin 2 θ+( cos 2 θ-i sin 2 θ) =2 cos 2 θ and when z= cos θ-i sin θ z2+z-2 = cos 2 θ-i sin 2 θ+ cos 2 θ+i sin 2 θ =2 cos 2 θ

Q. If $z^{2} - 2 z cos θ + 1 = 0$ , then $z^{2} + z^{- 2}$ is equal to

$2 cos 2 θ$

$2 sin 2 θ$

$2 cos θ$

$2 sin θ$

Q. If z2−2zcosθ+1=0, then z2+z−2 is equal to

2cos2θ

2sin2θ

2cosθ

2sinθ

We have, z2−2zcosθ+1=0 ⇒z=22cosθ±4cos2θ−4​​ =cosθ±cos2θ−1​ =cosθ±−sin2θ​ =cosθ±i2sin2θ​ =cosθ±isinθ When z=cosθ+isinθ z2+z−2=cos2θ+isin2θ+(cos2θ−isin2θ) =2cos2θ and when z=cosθ−isinθ z2+z−2=cos2θ−isin2θ+cos2θ+isin2θ =2cos2θ

Q. If $z^{2} - 2 z cos θ + 1 = 0$ , then $z^{2} + z^{- 2}$ is equal to

$2 cos 2 θ$

$2 sin 2 θ$

$2 cos θ$

$2 sin θ$