Sin Difference 817B24

1. **State the problem:** We want to prove the trigonometric identity $$\sin(x-y) = \sin x \cos y - \cos x \sin y.$$\n\n2. **Recall the sine difference formula:** The sine of a difference of two angles $x$ and $y$ is given by the formula $$\sin(x-y) = \sin x \cos y - \cos x \sin y.$$ This is a standard trigonometric identity derived from the unit circle or angle addition formulas.\n\n3. **Derivation using angle addition formula:** We know the sine addition formula: $$\sin(a+b) = \sin a \cos b + \cos a \sin b.$$\n\n4. Substitute $b = -y$ into the addition formula: $$\sin(x + (-y)) = \sin x \cos(-y) + \cos x \sin(-y).$$\n\n5. Use the even-odd properties of cosine and sine: $$\cos(-y) = \cos y,$$ $$\sin(-y) = -\sin y.$$\n\n6. Substitute these back: $$\sin(x-y) = \sin x \cos y + \cos x (-\sin y) = \sin x \cos y - \cos x \sin y.$$\n\n7. **Conclusion:** We have shown that $$\sin(x-y) = \sin x \cos y - \cos x \sin y,$$ which proves the identity.