Question

If $\sin x-\cos y=\sin \frac{\pi }{7}$ and $\cos x+\sin y=\cos \frac{\pi }{7}$ then find the value of $\sin (x-y)$.

Answer 1

Given $\sin x-\cos y=\sin \frac{\pi }{7}$......(1) and $\cos x+\sin y=\cos \frac{\pi }{7}$ .....(2).

Now squaring (1) and (2) and then adding we have,

$({\sin }^{2}x-2\sin x.\cos y+{\cos }^{2}y)+({\cos }^{2}x+2\cos x.\sin y+{\sin }^{2}y)={\sin }^2\frac{\pi }{7}+{\cos }^2\frac{\pi }{7}$

or, $2-2\sin (x-y)=1$

or, $2-1=2\sin (x-y)$

or, $\sin (x-y)=\frac{1}{2}$.