explain why cos (-3π/4) = cos (5π/4).

Explanation:

Step1: Recall cosine - function property

The cosine function is an even function, i.e., $\cos(-x)=\cos(x)$ for all real - valued $x$. So, $\cos\left(-\frac{3\pi}{4}\right)=\cos\left(\frac{3\pi}{4}\right)$.

Step2: Use angle relationship

We know that $\cos(x)=\cos(2\pi - x)$. Also, $\frac{5\pi}{4}=2\pi-\frac{3\pi}{4}$. So, $\cos\left(\frac{3\pi}{4}\right)=\cos\left(2\pi - \frac{3\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)$. Since $\cos\left(-\frac{3\pi}{4}\right)=\cos\left(\frac{3\pi}{4}\right)$ and $\cos\left(\frac{3\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)$, we have $\cos\left(-\frac{3\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)$.

Answer:

The cosine function is even ($\cos(-x)=\cos(x)$), so $\cos\left(-\frac{3\pi}{4}\right)=\cos\left(\frac{3\pi}{4}\right)$. And since $\cos(x)=\cos(2\pi - x)$ and $\frac{5\pi}{4}=2\pi-\frac{3\pi}{4}$, then $\cos\left(\frac{3\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)$, thus $\cos\left(-\frac{3\pi}{4}\right)=\cos\left(\frac{5\pi}{4}\right)$.