Algebra> answer this question with proper explanat...

answer this question with proper explanation Complex numbers

Janhavi , 7 Years ago

Grade 11

1 Answers

Rajdeep

Last Activity: 7 Years ago

HELLO THERE!

First, let’s look at the equation closely:

u² – 2u + 2 = 0

Alpha and Beta are the roots of the equation.

Sum of roots and product of roots are respectively:

$\alpha + \beta = 2\\ \alpha \beta = 2 \\ \\ (\alpha-\beta)^{2} = (\alpha+\beta)^{2} - 4\alpha \beta \\ \implies \alpha-\beta = \sqrt{-4} = 2i$

Now on solving the two equations:

$\alpha +\beta =2\\ \alpha -\beta = 2i$

We get the roots as:

$\alpha = 1+i\\ \beta = 1-i$

Now, we are given that cot theta = x + 1,

$x = cot \theta - 1$

Now in the given equation, put the value of x, alpha and beta:

$\frac{(x+\alpha)^{n} -(x+\beta)^{n}}{\alpha-\beta} \\= \frac{(cot\theta - 1+1+i)^{n}-(cot\theta-1+1-i)^{n}}{2i} \\= \frac{(cot\theta+i)^{n}-(cot\theta-i)^{n}}{2i}$

Now, put:

$cot\theta = \frac{cos\theta}{sin\theta}$

The equation now becomes:

$\frac{(e^{i\theta})^{n} -(e^{-i\theta})^{n}}{2i\times sin^{n}\theta} \\\\=\frac{cosn\theta+isinn\theta-cosn\theta+isinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{2i\times sinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{sinn\theta}{sin^{n}\theta}$