Answer
True
Work Step by Step
Since the period of cosine is $2\pi$ it follows:
$\cos(2\pi+\frac{\pi}{7})=\cos(\frac{\pi}{7})$
$\cos^{2}(2\pi+\frac{\pi}{7})=\cos^{2}(\frac{\pi}{7})$
so:
Replace $\cos^{2}\left(2\pi+\frac{\pi}{7})\right)$ by $\cos^2\left(\frac{\pi}{7}\right)$ in the expression:
$\sin^{2}(\frac{\pi}{7})+\cos^{2}(2\pi+\frac{\pi}{7})=\sin^{2}(\frac{\pi}{7})+\cos^{2}(\frac{\pi}{7})$.
Use the Pythagoreic identity:
$\cos^2\theta+\sin^2\theta=1$
$\sin^{2}(\frac{\pi}{7})+\cos^{2}(\frac{\pi}{7})=1$
The given equation is true.
