Chapter 4.3, Problem 38E

Prove the sum of angles formula for the sine function by following these steps. Fix $x$

a) Let $f\left(t\right):\equiv sin\left(x+t\right).$ Show that $f^{″}\left(t\right)+f\left(t\right)=0$, $f\left(0\right)=\sin x$, $f^{\prime }\left(0\right)=\cos x$.

b) Use the auxiliary equation technique to solve the initial value problem $y^{″}+y=0$, $y\left(0\right)=\sin x$, and $y^{\prime }\left(0\right)=\cos x$.

c) By uniqueness, the solution in part (b) is the same as $f\left(t\right)$ from part (a).Write this equality; this should be the standard sum of angles formula for $\sin \left(x+t\right)$.