Suppose that f (x) satisfies the following equation (an example of a differential equation):
f″(x) = −f (x)
(a) Show that f (x)2 + f′(x)2 = f (0)2 + f′(0)2 for all x.
(b) Verify that sin x and cos x satisfy Eq. (1), and deduce that sin2 x + cos2 x = 1.