If $y = \cos \theta + i\sin \theta$,then the value of $y + \frac{1}{y}$ is

If $n$ is a positive integer,then $(1+i \sqrt{3})^n+(1-i \sqrt{3})^n$ is equal to

The sum of the products of the non-conjugate roots of $i^{1/4}$ taken two at a time is

If $\omega$ is an imaginary cube root of unity,then the value of $(1-\omega+\omega^{2}) \cdot(1-\omega^{2}+\omega^{4}) \cdot(1-\omega^{4}+\omega^{8}) \cdot \ldots$ ($2n$ factors) is

$\sum_{n=1}^{20} \left[ \sin \left( \frac{2n\pi}{21} \right) - i \cos \left( \frac{2n\pi}{21} \right) \right] = $

If $\alpha, \beta$ are the roots of the equation $x^2-2x+4=0$ and for any $n \in N, \alpha^n+\beta^n=k \cos \frac{n \pi}{3}$, then $k=$