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        answer this question with proper explanation Complex numbers
one year ago

Rajdeep
231 Points
							HELLO THERE! First, let’s look at the equation closely: u2 – 2u + 2 = 0Alpha 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,so $x = cot \theta - 1$ 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: $\frac{(e^{i\theta})^{n} -(e^{-i\theta})^{n}}{2i\times sin^{n}\theta} \\\\=\frac{cosn\theta+isinn\theta-cosn\theta+isinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{2i\times sinn\theta}{2i\times sin^{n}\theta} \\\\= \frac{sinn\theta}{sin^{n}\theta}$ So, your answer is option (A).The explanation is above. THANKS!

one year ago
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