Q.Show that $\left(\dfrac{\sqrt3}2+\dfrac i2\right)^5+\left(\dfrac{\sqrt3}2-\dfrac i2\right)^5=-\sqrt3$.

de Moivre's Theorem. For any complex number $\cos\theta+i\sin\theta$ (modulus $1$) and any integer $n$,

In Euler form this is simply the exponent law $\left(e^{i\theta}\right)^n=e^{in\theta}$. It converts a power of a trigonometric expression into a single trigonometric expression at a multiplied angle — the tool that carries trigonometry "out of geometry and into analysis."

Corollaries (each following by substituting $-\theta$ or $-n$, or both, into the theorem):

1. $(\cos\theta-i\sin\theta)^n=\cos n\theta-i\sin n\theta$
2. $(\cos\theta+i\sin\theta)^{-n}=\cos n\theta-i\sin n\theta$
3. $(\cos\theta-i\sin\theta)^{-n}=\cos n\theta+i\sin n\theta$
4. $\sin\theta+i\cos\theta=i(\cos\theta-i\sin\theta)$

Working method for a general complex number. To raise $z=x+iy$ to a power $n$: (1) convert $z$ to polar form $r(\cos\theta+i\sin\theta)$ (find $r=|z|$ and the correct-quadrant $\theta$); (2) apply de Moivre's theorem to get $r^n(\cos n\theta+i\sin n\theta)$; (3) reduce the angle $n\theta$ modulo $2\pi$ (or $360°$) to a standard range and convert back to rectangular form if required.

Common applications:

• Simplifying a high power directly, e.g. $(1+i)^{18}$: convert $1+i$ to $\sqrt2\operatorname{cis}(\pi/4)$, then $(1+i)^{18}=2^9\operatorname{cis}(18\pi/4)=2^9\operatorname{cis}(\pi/2)=512i$.
• Combining conjugate-style powers: if $z=\cos\theta+i\sin\theta$, then $z^n+z^{-n}=2\cos n\theta$ and $z^n-z^{-n}=2i\sin n\theta$ — the algebraic route between powers of $\cos\theta+i\sin\theta$ and multiple-angle trig identities; the substitution $2\cos\alpha=x+1/x$ (so $x=\cos\alpha+i\sin\alpha$) is the general version of this trick.
• Rotations: multiplying $z$ by $e^{i\theta}$ rotates $z$ by $\theta$ radians counter-clockwise about the origin (a direct reading of $z\,e^{i\theta}=|z|e^{i(\arg z+\theta)}$) — this is how a vertex of a regular polygon inscribed in a circle is obtained from another vertex.
• Argument of a power: since $\arg(z^n)=n\arg z$, de Moivre's theorem is also the fastest route to the principal argument of an expression like $(\sin40°+i\cos40°)^5$ (first rewrite $\sin\theta+i\cos\theta$ as $\cos(90°-\theta)+i\sin(90°-\theta)$, then apply the theorem).