Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 46

$$\sin(\pi+x)=-\sin x$$

$$X=\sin(\pi+x)$$ According to the identity of the sum of sines: $$\sin(A+B)=\sin A\cos B+\cos A\sin B$$ Expand $X$: $$X=\sin\pi\cos x+\cos\pi\sin x$$ $$X=0\cos x+(-1)\sin x$$ $$X=-\sin x$$ Overall, $$\sin(\pi+x)=-\sin x$$