ANSWER and EXPLANATION
We want to prove that:
[tex]\sin(x+\pi)=-\sin x[/tex]
Let us start with the left-hand side of the equation.
Using trigonometric identities for sine, we have that:
[tex]\sin(A+B)=\sin A\cos B+\sin B\cos A[/tex]
Applying this identity to the left-hand side of the equation:
[tex]\sin(x+\pi)=\sin x\cos\pi+\sin\pi\cos x[/tex]
We know that:
[tex]\begin{gathered} \cos\pi=-1 \\ \sin\pi=0 \end{gathered}[/tex]
Substituting those values into the above expression:
[tex]\begin{gathered} \sin x(-1)+(0)\cos x \\ \Rightarrow-\sin x \end{gathered}[/tex]
Since the left-hand side of the equation is equal to the right-hand side, we have that it has been proven.
