Prove the sum of angle formula for the sine function by following these steps. Fix x.
(a) Let f(t):= sin (x + t). Show that f’’(t) + f(t) = 0, f(0) = sin x, and f ‘(0)= cos x.
(b) Use the auxiliary equation technique to solve the initial value problem y’’+ y = 0, y(0) = sin x, and y’(0)= cos x.
(c) By uniqueness, the solution in part (b) is the same as f (t) from part (a). Write this equality; this should be the standard sum of angle formula for sin (x + t).
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