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Complex numbers: powers of cis θ and integration

mathematical

Let

z = \cos\theta + i\sin\theta

Facts you may need

$\frac1z = \text{cis}(-\theta)$ when $|z| = 1$ — So $z^n + z^{-n} = 2\cos n\theta$.
Binomial coefficients for power 4: $1, 4, 6, 4, 1$ — Pascal's triangle.
$\int_0^{\pi/2}\cos 2\theta\,d\theta = \int_0^{\pi/2}\cos 4\theta\,d\theta = 0$ — Full half-periods cancel.

Show that

z^n + z^{-n} = 2\cos n\theta

for positive integers

n

By expanding

\left(z + \frac1z\right)^4

, show that

\cos^4\theta = \frac18\left(\cos 4\theta + 4\cos 2\theta + 3\right)

Hence find the exact value of

\displaystyle\int_0^{\pi/2} \cos^4\theta\, d\theta