91Ó°ÊÓ

Establish the orthogonality of the trigonometric system over the interval \([-\pi, \pi]\).

(a) Use Parseval's identity and the Fourier series expansion $$ \frac{x}{2}=\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin n x, \quad-\pi

Find the sine series expansion of the given function on the interval \(0

Find the sine series expansion of the given function on the interval \(0

(a) Prove De Moivre's identity which states that $$ \cos n \theta+i \sin n \theta=(\cos \theta+i \sin \theta)^{n} $$ [Hint: Use Euler's identity and basic properties of the exponential function. (b) Use De Moivre's identity with \(n=2\) to show that $$ \cos 2 \theta=\cos ^{2} \theta-\sin ^{2} \theta \text { and } \sin 2 \theta=2 \sin \theta \cos \theta $$ (c) Derive the identities: \(\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta\) and \(\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta\).