If x satisfies x2-2x text cosθ +1=0, then the value of ( xn+(1/xn) ) is:

Question

If x satisfies x2-2x text cosθ +1=0, then the value of ( xn+(1/xn) ) is: (A)  2n cos nθ  (B)  2n cos nθ  (C)  2 cos nθ  (D)  2 cos nθ

Tardigrade Admin · Accepted Answer

x2-2x cos θ +1=0 ⇒ x=(2 cos θ ± √4 cos 2θ -4/2) x= cos θ ± i sin θ Take + sign, x= cos θ +i sin θ ⇒ xn= cos nθ +i sin nθ ∴ xn+(1/xn)= cos nθ +i sin nθ + cos nθ -i sin nθ =2 cos nθ

Q. If $x$ satisfies $x^{2} - 2 x cos θ + 1 = 0,$ then the value of $(x^{n} + \frac{1}{x ^{n}})$ is:

$2^{n} cos n θ$

$2^{n} cos^{n} θ$

$2 cos^{n} θ$

$2 cos n θ$

$2 sin n θ$

$x^{2} - 2 x cos θ + 1 = 0$ $\Rightarrow$ $x = \frac{2 c o s θ \pm 4 c o s ^{2} θ - 4}{2}$ $x = cos θ \pm i sin θ$ Take $+$ sign, $x = cos θ + i sin θ$ $\Rightarrow$ $x^{n} = cos n θ + i sin n θ$ $∴$ $x^{n} + \frac{1}{x ^{n}} = cos n θ + i sin n θ + cos n θ - i sin n θ$ $= 2 cos n θ$

Q. If x satisfies x2−2xcosθ+1=0, then the value of (xn+xn1​) is:

2ncosnθ