Why is this true? $\cos\left(\left(n+ \frac{1}{2}\right)\pi\right)=\cos\left(\frac{1}{2} \pi\right) = 0$

Question

Why is the following true? $$ \cos\left(\left(n+ \frac{1}{2}\right)\pi\right)=\cos\left(\frac{1}{2} \pi\right) = 0 $$

Answer 1

Because $\cos(\pi+x)=-\cos(x)$. If $\cos(\pi/2)=0$, then $\cos(\pi/2+\pi)=-0=0$, $\cos(\pi/2+2\pi)=\cos(\pi/2)=0$. For the last one I've used the periodicity of the cosine function. All the rest of the terms you get from periodicity as well.

Answer 2

Because introducing the value $n$ shifts the cosine function by a multiple of $\pi$. Since the cosine function has a zero at $\frac{\pi}{2}$, and zeros occur every $\pi$ units, this means that your expression is always $0$.