If x + frac{1}{x} = 2cos alpha,then x^n + fra | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given $x + \frac{1}{x} = 2 \cos \alpha$. Let $x = \cos \alpha + i \sin \alpha = e^{i\alpha}$. Then $\frac{1}{x} = \cos \alpha - i \sin \alpha = e^

Vedclass · Accepted Answer

$2 \cos n\alpha$

Vedclass · Answer

(D) Given $x + \frac{1}{x} = 2 \cos \alpha$. Let $x = \cos \alpha + i \sin \alpha = e^{i\alpha}$. Then $\frac{1}{x} = \cos \alpha - i \sin \alpha = e^{-i\alpha}$. Thus,$x + \frac{1}{x} = e^{i\alpha} + e^{-i\alpha} = 2 \cos \alpha$. Now,$x^n + \frac{1}{x^n} = (e^{i\alpha})^n + (e^{-i\alpha})^n = e^{in\alpha} + e^{-in\alpha}$. Using Euler's formula,$e^{in\alpha} + e^{-in\alpha} = 2 \cos n\alpha$. Therefore,$x^n + \frac{1}{x^n} = 2 \cos n\alpha$.

If $x + \frac{1}{x} = 2\cos \alpha$,then $x^n + \frac{1}{x^n} = $

Similar Questions

Express $\frac{(\cos 2\theta - i\sin 2\theta)^4 (\cos 4\theta + i\sin 4\theta)^{-5}}{(\cos 3\theta + i\sin 3\theta)^{-2} (\cos 3\theta - i\sin 3\theta)^{-9}}$ in the form $x + iy$.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-2x+4=0$,then $\alpha^{12}+\beta^{12}=$

If $\alpha$ and $\beta$ are the roots of the equation $x^2+x+1=0$,then the equation whose roots are $\alpha^{19}$ and $\beta^7$ is

One of the roots of the equation $(x+1)^4 + 81 = 0$ is

If $\alpha$ and $\beta$ are the roots of $x^{2}+x+1=0$,then $\alpha^{16}+\beta^{16}$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module