Practice (116)

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Let $x, y \in \big(0, \frac{\pi}{2}\big)$. Show that if the equation $(\cos x + i \sin y)^n = \cos nx + i \sin ny$ holds for two consecutive positive integers, then it will hold for all positive integers.

Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$

Let $A (x_1, y_1)$, $B (x_2, y_2)$, and $C (x_3, y_3)$ be three points on the unit circle, and $$x_1 + x_2 + x_3 = y_1+y_2+y_3=0$$ Prove $$x_1^2 +x_2^2+x_3^2=y_1^2+y_2^2+y_3^2=\frac{3}{2}$$

Let $z=\cos{\theta} + i\sin{\theta} $. Show $z^{-1} = \cos{\theta} - i\sin{\theta}$.