prove the equation using complex numbers

Question

Let $\alpha, \beta$ be the root of the equation $t^2-2t+5=0$ where $n$ is a positive integer. Prove that $$\frac{(a+\alpha)^n-(a+\beta)^n}{(\alpha-\beta)} = 2^{n-1} \sin( n\phi) \csc^{n} (\phi),$$ where $a$ is a real number satisfying $\frac{1}{2}(a+1)= \cot{\phi}$

Answer 1

Calculate the roots of the equation as $1+2i,1-2i$

Now put the given values on the L.H.S,

$$\frac{(2\cot{\phi}-1+1+2i)^n-(2\cot{\phi}-1+1-2i)^n}{4i}$$

$$\frac{(2\cot{\phi}+2i)^n-(2\cot{\phi}-2i)^n}{4i}$$

$$\frac{2^{n-2}((\cot{\phi}+i)^n-(\cot{\phi}-i)^n)}{i}$$

$$\frac{2^{n-2}\csc^n{\phi}((\cos{\phi}+i\sin{\phi})^n-(\cos{\phi}-i\sin{\phi})^n)}{i}$$

Using De-Movires Theorem,

$$2^{n-1}\csc^n{\phi}\sin{n\phi}$$