Show that $\sin\left(\frac\pi3(x-2)\right)$ is equal to $\cos\left(\frac\pi3(x-7/2)\right)$
The equality $\sin \alpha=\cos\beta$ can be written $$ \cos(\pi/2-\alpha)=\cos\beta $$ which is satisfied when either $$ \beta=\frac{\pi}{2}-\alpha+2k\pi \qquad(k \text{ integer}) $$ or $$ \beta=\alpha-\frac{\pi}{2}+2k\pi \qquad(k \text{ integer}) $$ These can be rewritten respectively as $$ \alpha+\beta=\frac{\pi}{2}+2k\pi \qquad(k \text{ integer}) $$ or $$ \alpha-\beta=\frac{\pi}{2}+2k\pi \qquad(k \text{ integer}) $$ Now try with $\alpha=(\pi/3)(x-2)$ and $\beta=(\pi/3)(x-7/2)$; is one of the two equalities true?